FEASIBILITY STUDY OF A SYNTHESIS PROCEDURE FOR

ARRAY FEEDS TO IMPROVE RADIATION PERFORMANCE OF

LARGE DISTORTED REFLECTOR ANTENNAS

Grant No. NAG-1-859

SEMIANNUAL STATUS REPORT

by

W. L. Stutzman

W. T. Smith

Virginia Polytechnic Institute and State University

Bradley Department of Electrical Engineering

Blacksburg, VA 24061

Virginia Tech Report

EE SATCOM 90-2

August, 1990

Table of Contents

1.0 INTRODUCTION ..................................................... I

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES ...... 6

2.1 REFLECTOR SURFACE ERRORS ..................................... 8

2.1.1 RANDOM SURFACE ERRORS ..................................... 8

EARLY WORK FOR UNIFORM AMPLITUDE APERTURES ........... 8

TAPERED APERTURE DISTRIBUTION RESULTS .................. 10

GENERAL ROUGH SURFACE EFFECTS ON THE RADIATION

PATTERN .................................................. 14

2.1.2 RADIAL SURFACE ERRORS ...................................... 15

2.1.3 AZIMUTHAL SURFACE ERRORS ................................. 19

PARAMETRIC STUDY OF TttE EFFECTS OF THE AZIMUTHAL

ERRORS CAUSED BY THE RADIAL RIB REFLECTOR ............. 23

2.1.4 RADIAL AND AZIMUTHAL SURFACE ERRORS ..................... 23

2.1.5 SUMMARY OF SURFACE ERROR EFFECTS ........................ 24

2.2 FOCAL REGION FIELDS ............................................ 26

2.2.1 FOCAL REGION ANALYSIS OF SMOOTH AND ROUGH REFLECTORS .. 27

Table of Contents i J

AXISYMMETRIC REFLECTORS ................................ 27

OFFSET REFLECTORS ........................................ 36

2.2.2 FOCAL REGION ANALYSIS OF A SCANNED REFLECTOR ............ 36

SCANNED FOCAL REGION FIELDS, SMOOTH REFLECTOR ........ 36

SCANNED FOCAL REGION FIELDS, ROUGH REFLECTOR ........ 38

2.3 PAST ELECTROMAGNETIC COMPENSATION TECHNIQUES ............. 43

2.3.1 DOUBLE FOURIER TRANSFORM TECHNIQUE ..................... 43

2.3.2 CONJUGATE FIELD MATCH TECHNIQUE (COMPUTATIONAL) ...... 49

CONJUGATE FIELD MATCH BACKGROUND ..................... 50

CFM COMPENSATION ........................................ 55

ICFM AND MODIFIED ICFM WITH IMPROVED SIDE LOBE

PERFORMANCE ............................................ 55

DCFM WITH IMPROVED SIDE LOBE PERFORMANCE ............. 61

2.3.3 LEAST SQUARED ERROR TECHNIQUE ............................ 66

2.3.4 CONJUGATE FIELD MATCH TECHNIQUE (EXPERIMENTAL) ......... 76

2.3.5 SUMMARY OF COMPENSATION TECHNIQUES ..................... 80

3.0 REFLECTOR SCAN_ING ............................................ 82

3.1 DISPLACED FEED SCANNING ....................................... 82

3.1.1 PHASE ERROR TYPES .......................................... 85

3.1.2 BEAM DEVIATION FACTOR FOR AXISYMMETRIC REFLI'CTORS ..... 91

3.1.3 BEAM DEVIATION FACTOR FOR OFFSET REFLECTORS ............. 98

3.2 ELECTRONIC SCANNING ........................................... 98

3.2.1 FIXED ARRAY FEED SCANNING .................... , ........... I00

3.2.2 MOVABLE FEED ARRAY SCANNING ............................. 104

3.3 DUAL REFLECTOR SCANNING ..................................... 112

Table of Contents iii

4.0 PATTERN SYNTHESIS COMPENSATION TECHNIQUE .................. 120

4.1 ITERATIVE SAMPLING METHOD ................................... 121

4.1.1 ITERATIVE SAMPLING METHOD ALGORITHM .................... 121

4.1.2 APPLICATION OF ISM TO REFLECTORS .......................... 123

4.2 PATTERN SYNTHESIS COMPENSATION TECHNIQUE DESCRIPTION ..... 129

5.0 COMPUTER CODES ............................................... 135

5.1 REFLECTOR ANALYSIS PROGRAM FOR CYLINDRICAL REFLECTORS

(RAPCA) ........................................................... 136

5.1.1 RAPCA ALGORITHM ........................................... I36

5.1.2 RAPCA INPUT AND OUTPUT ................................... 142

5.2 SCANNED REFLECTOR ANALYSIS PROGRAM (SCANRAP) ............. 143

5.2.1 SCANRAP ALGORITtlM ........................................ 143

5.2.2 SCANRAP INPUT AND OUTPUT ................................. 155

5.3 DUAL REFLECTOR ANALYSIS PROGRAM FOR CYLINDRICAL

REFLECTORS (DRAPCA) ............................................. 155

5.3.1 DRAPCA ALGORITHM ......................................... 155

DERIVATION OF THEUNNORMALIZED RADIATION INTEGRAL .. 160

E-FIELD EXPRESSIONS FOR PO CURRENT CALCULATIONS ...... 163

5.3.2 DRAPCA INPUT AND OUTPUT .................................. 164

5.4 REFLECTOR ANALYSIS PROGRAM (RAP), VERSION 2.0 ................ 165

5.4.1 RAP ALGORITHM ............................................. 165

5.4.2 RAP INPUT AND OUTPUT ...................................... 173

6.0 COMPENSATION RESULTS ........................................ 174

6.1 SCANNING STUDIES .............................................. 175

6.1.1 PRIME FOCUS CYLINDRICAL REFLECTOR SCANNING STUDY ...... 175

FEED DISPLACEMENT ...................................... 176

Table of Contents JV

ELECTRONIC SCANNING .................................... 182

6.1.2 DUAL REFLECTOR SCANNING STUDIES ......................... 186

6.2 CYLINDRICAL REFLECTOR COMPENSATION ANALYSIS .............. 192

6.2.1 COMPENSATION WITH AN AUXILIARY FEED ..................... 192

EXTENDED RANGE COMPENSATION ......................... 196

SIDE LOBE PHASE DETERMINATION .......................... 196

6.2.2 ELECTRONIC COMPENSATION ................................. 198

6.3 PARABOLOIDAL REFLECTOR COMPENSATION ANALYSIS ............. 205

6.3.1 COMPENSATION WITH AUXILIARY DISPLACED FEEDS ............ 206

6.3.2 ELECTRONIC COMPENSATION ................................. 213

6.4 DUAL REFLECTOR COMPENSATION ANALYSIS ...................... 216

7.0 SUMMARY AND CONCLUSIONS .................................... 222

8.0 REFERENCES .................................................... 226

9. APPENDIX A: EXAMPLE COMPUTER INPUTS AND OUTPUTS ............ 231

A.1 RAPCA INPUT AND OUTPUT ...................................... 231

A.2 SCANRAP INPUT AND OUTPUT ................................... 238

A.3 DRAPCA INPUT AND OUTPUT .................................... 242

Table of Contents V

1.0 INTRODUCTION

Surface errors on parabolic reflector antennas degrade the overall performance of

the antenna. The errors are in the form of roughness on the surface, deviations of the

mean surface from the true parabolic shape, or structural design details such as ribbing,

slots between panels, etc.. They cause amplitude and phase errors in the aperture field

which lower the gain, raise the side lobes, and fill in the nulls. These are major problems

in large space reflector antenna systems. Planned mobile satellite communications

systems having limited signal margin need high gain from the space reflectors. Future

multiple beam antenna systems requiring spatial isolation to allow frequency reuse could

be rendered useless if high side lobes are present. Planned remote sensing missions, such

as Mission to Planet Earth and Global Change Technology Initiative, require high beam

efficiencies. High side lobes are responsible for noise in remote sensing systems.

Space antenna structures are difficult to build, deploy and control. They must

maintain a nearly perfect parabolic shape in a harsh environment and must be

lightweight. The restrictions on the structure become more severe as science and

technology requirements demand electrically large antennas. There are technologies for

building antennas with mechanically adaptive surfaces that can compensate for many

1.0 INTRODUCTION
of the larger distortions caused by thermal and gravitational forces. The NASA

Hoop/Column 15-meter mesh reflector uses a series of cables to adjust the surface profile

to have an error of less than 70 mils. However, as the frequency and size of the

reflectors increase, the subtle surface errors become significant and degrade the overall

radiation pattern. It is for this reason that another method must be used to further

improve the radiation pattern.

Electromagnetic compensation for surface errors in large space reflector antennas

can be used to supplement mechanical compensation. In order to implement

electromagnetic compensation, some information about the reflector surface or the

electromagnetic fields (focal region fields or the radiated far-fields) are required. If

accurate surface data are available, the fields information can be computed using

diffraction analysis.

Measurements of the necessary data for performing electromagnetic compensation

are difficult to obtain. Current methods for accurate measurement of the reflector

surface on earth include optical metrology [22] and microwave holography metrology

[35]. There are many methods for measuring the radiated far-fields on earth. A

technique for determining the far-fields of a space reflector using a free flying probe has

been proposed. [44] Recently, focal region field data were measured using the array feed

system. [43]

Electromagnetic compensation for surface errors in large space reflector antennas

has been the topic of several research studies. Most of these studies try to correct the

focal plane fields of the reflector near the focal point and, hence, compensate for the

distortions of the over the whole radiation pattern. The compensation is implemented

by weighting the elements of an array feed. Of course, perfect compensation would

require a very large array. In most of the studies, a precise knowledge of the reflector

surface is required (surface shape and the first and second derivatives). At the present,

!.0 INTRODUCTION 2
this approach works on earth but the sensing of the surface errors in space is an

especially difficult problem.

An alternative approach to electromagnetic compensation is presented in this

study. The proposed technique uses pattern synthesis to compensate for the surface

errors. It differs from previous methods in two major respects. First, the previous

studies used a global algorithm that tries to correct the entire focal plane field near the

focal point and modify the entire radiation pattern. The new pattern synthesis approach

uses a localized algorithm in which pattern corrections are directed specifically towards

portions of the pattern requiring improvement. The second major difference is that the

pattern synthesis technique does not require knowledge of the reflector surface. It uses

radiation pattern data to perform the compensation.

Chapters 2 and 3 of this report contain necessary background information required

to implement the pattern synthesis compensation technique and discuss past

electromagnetic compensation studies. In Section 2.1 the effccts of different types of

surface errors are analyzed. The information on the effects of surface errors helps to

define what types of distorted reflector patterns can be improved with electromagnetic

means. Section 2.2 provides insight into what happens to the focal plane fields when the

reflector is distorted and scanned. This information is necessary when trb'ing to design

an array feed that sufficiently samples the focal plane fields. Section 2.3 reviews past

electromagnetic compensation efforts. The kinds of reflector data required to implement

the techniques are outlined and the limitations of the methods are noted.

Chapter 3 contains background information on reflector scanning techniques. This

information was separated from the discussion in Ch. 2 to emphasize the importance of

reflector scanning to the pattern synthesis compensation technique. Section 3.1

evaluates scanning of a reflector by displacing the feed. Section 3.2 examines electronic

scanning of a reflector with an array feed. Section 3.3 deals with dual reflector scanning.

1.0 INTRODUCTION
All of the above scanning techniques were used to implement the pattern synthesis

compensation technique (see Ch. 6).

Chapter 4 presents the pattern synthesis compensation technique algorithm. The

technique is based on the iterative sampling method (ISM). In Section 4.1, the ISM is

formulated to apply to reflector antennas. Section 4.2 discusses the compensation

algorithm and its limitations.

New and existing computer codes were used to analyze reflector scanning and

implement the pattern synthesis compensation technique. Chapter 5 contains algorithm

descriptions. Section 5.1 discusses the Reflector Analysis Program for Cylindrical

Antennas (RAPCA). RAPCA is an infinite cylindrical prime focus reflector code.

Patterns with single or array feeds can be computed. The SCANned Reflector Analysis

Program (SCANRAP) is discussed in Section 5.2. SCANRAP is used to determine the

array feed excitations for electronic scanning of infinite cylindrical prime focus reflector

antennas. The SCANRAP excitations are fed into RAPCA to calculate the

electronically scanned pattern. The Dual Reflector Analysis Program for Cylindrical

Antennas (DRAPCA) is capable of analyzing infinite cylindrical dual reflector antennas

with symmetric or offset geometries and single or array feeds. Section 5.3 presents the

theory behind the dual reflector analysis and describes the problem geometry. The

Reflector Analysis Program (RAP), Version 2.0, is an existing prime focus paraboloidal

reflector code. RAP analyzes single feed reflectors with axisymmetric or offset

geometries. It was also used to analyze paraboloidal reflectors with array feeds by

generating patterns for the individual elements and then superposing the patterns. RAP

is outlined in Section 5.4.

The results of this study are contained in Chapter 6. Section 6.1 presents various

reflector scanning techniques that are used to implement the pattern synthesis

compensation technique. Particular attention is paid to the main beam characteristics

1.0 INTRODUCTION
of distorted reflectors. Sections 6.2, 6.3, and 6.4 present several compensation problems

for infinite cylindrical prime focus reflectors, prime focus paraboloidal reflectors, and

infinite cylindrical dual reflectors, respectively.

From the results in Chapter 6, the pattern synthesis compensation technique was

shown to be a feasible method to compensate for pattern irregularities in distorted

reflector antennas. The summary and conclusions of this study are presented in Chapter

7.

!.0 INTRODUCTION
2.0 REFLECTOR SURFACE ERRORS AND

COMPENSATION TECHNIQUES

Surface errors on parabolic reflector antennas degrade the overall performance of

the antenna. The errors are in the form of roughness on the surface, deviation of the

mean surface from the true parabolic shape, or structural design details (such as ribbing,

slots between panels, etc.). They cause amplitude and phase errors in the aperture field

which lower the gain, raise the side lobes, and fill in the nulls. These are maior problems

in large space reflector antenna systems. Planned mobile satellite communications

systems having limited signal margin need high gain from the space reflectors. Future

multiple beam antenna systems requiring spatial isolation to allow frequency reuse could

be rendered useless if high side lobes are present. High side lobes are also responsible

for noise. This is especially troublesome for radiometric systems such as in remote

sensing applications.

Methods for reducing the distortion of the aperture fields due to reflector surface

roughness include:

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES
Direct (mechanical adjustment)

Indirect (electromagnetic compensation)

Hybrid (a combination of both)

Mechanical means of correction for surface roughness typically involves first the

sensing of the surface shape and then an adjustment of the main reflector structure to

bring it back into shape at selected points. This approach may work well on earth but

is difficult in space. Sensing of surface errors is especially difficult in space. Also

mechanical correction devices such as motor actuators add weight and complexity to the

antenna.

Electromagnetic compensation has been investigated for nearly twenty years. Most

methods try to correct the amplitude and phase errors in the aperture field. The majority

of work in electromagnetic compensation for distorted reflectors involves the method of

conjugate field match (CFM). The CFM method has also been used to scan the main

beam of reflectors (See. 3.2). The technique uses a multi-element feed array to

compensate for amplitude and phase errors in the focal plane field distribution caused

by a distorted reflector surface.

Previous investigations into surface error types found in large space reflectors are

reviewed in Sec. 2.1. Both random and deterministic errors are discussed. The effects

of surface errors and scanning on the focal region fields are evaluated in Sec. 2.2. To

perform compensation with an array feed, one needs to adequately sample the focal

region fields. Feed location and array feed geometries depend on the distribution of the

fields. Section 2.3 is devoted to CFM and other related compensation techniques. The

theory behind the previous compensation methods is important when comparing them

to the technique proposed in this effort.

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES
2.1 REFLECTOR SURFACE ERRORS

2.1.1 RANDOM SURFACE ERRORS

The available theories concerning random surface errors treat reflectors with rough

surfaces (i.e. the surface is not perfectly smooth) using statistical characterizations of the

effects of the random errors. Expressions for the gain, SLL, etc. are derived in a

"mean" sense.

EARL Y WORK FOR UNIFORM AMPLITUDE APER TURES

Perhaps the most referenced work on the topic was written by Ruze. [29] The

paper deals with gain reduction due to scattering. Two important results are an

expression for the gain of a rough reflector and the frequency dependence of that gain.

The expressions are valid for surfaces with small correlation intervals. [30] The aperture

illumination is assumed nearly constant over distances on the order of the correlation

length. The gain of a rough reflector, G, is given by [29]

G _.Goe (2.1-I)

where Go is the gain of the corresponding perfectly smooth reflector given by [29]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES
where _/is the aperture efficiency of the reflector, D is the diameter of the aperture, and

_l is the wavelength. The exponent in (2.1-1) is [19]

32=( )2 (2.1-3)

,t

where _ is the rms surface error. An approximate rule is that the rms error is about

one-third of the peak deviation. The factor _ depends on the reflector geometry and is

approximately [ 19]

_: #" -'D-'4F4in[ 1 + I](4F]D)2] (2.1-4)

where F is the focal length of the reflector. For large F/D ratios, _: is nearly unity, and

using (2.1-2) and (2.1-3) in (2.1-I) gives the classic result for a circular aperture [29]

G "-~_/( _ )2 e-(4'_1;02 for large F/D . (2.1-5)

This expression shows the expected result that the gain of a rough reflector (e > 0) is

lower than that of a smooth reflector (_ = 0).

The frequency dependence of the gain of a rough reflector may be found from

(2.1-5) (assuming a constant r/). Initially the gain G increases with increasing frequency

(decreasing wavelength). For a certain wavelength ,t,_u, the effects of the surface errors

(the negative exponential term) cause the gain to begin to decrease with further

frequency increase due to the roughness. The wavelength at which maximum gain

occurs (based on (2.1-5)) is [291

,lm_ = 47r_ (2.1-6)

and the corresponding maximum gain value is approximately [291

2.0 REFLECTOR SURFACE ERRORS #,aND COMPENSATION TECHNIQUES
Gma, _,,._.( ._. )2 (2.1-7).

TAPERED APER TURE DISTRIBUTION RESULTS

Rahmat-Samii developed a model based on work by Ruze and others that evaluates

the average power pattern degradations caused by random surface errors. [301 The

model is somewhat more general than Ruze's in that it can handle tapered aperture

amplitude distributions and non-uniform rms distortions. The geometry of the model is

shown in Fig. 2.1-1. It is assumed that the rms surface error is known in each annular

region. The aperture field amplitude distribution is assumed to be of the parabola on a

pedestal shape: [301

Q(p' /9'2 )P ) = B + C(l 2 (2.1-8)

a

where B + C = 1, a is the radius of the aperture, and p = 1 or 2. The parameter p is

used to control the shape of the aperture distribution.

Some important conclusions from Rahmat-Samii's work concern the gain loss and

the side lobe degradation. The gain loss correction factor x in (2.1-4) was computed for

different F]D ratios and different edge tapers as a function of _/,_ . The results are

plotted in Fig. 2.1-2. The general trend shows that as the F/D decreases, so does _:.

Decreasing _: means that the surface errors have decreasing impact on the gain loss of

the antenna. It was also found in the study that _: is relatively insensitive to variations

in edge taper for given FID and _/A. [30]

Representative plots for side lobe level increase and main beam peak loss versus

surface roughness are shown in Fig. 2.1-3. The general trend is for the side lobe levels

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 10
Y

l ,ith ZONE IN THE

/nth ANNULAR REGION

___ / (i = 1 TO Kn)

aN--O _ _x

Figure 2.1-1. Reflector geometry model showing the aperture divided into N annular

rings. The annular rings are subdivided into K, zones in the nth ring

(n ffiltoN). [301

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES

11
1.0

0.9tJ77

0.9758

0.9638

0,9427

0.9209

0.90

O. 8799

0.80

O,_FO _

I/lO0 I/SO

I " 1

I/iS

I • I

1/20

I " I

.ram

I/tO

| ' I " -

0.9723

O.9607

O.9492

O.929O 0.5

0.9078 0.4

(EDGE TARR • -10 rill) 0.3

0.8681

(o dB) 0.3

#'G • 10 log 41_41r_lrrlm/ll )2

. I • I • I 1 ! • ! . ! , | 1

0.01 0.02 0,03 0.04 0.05 0,06 0.07 0._ 0.09 0.

%m/k

Figure 2.1-2. Plots of the correction factor _cas a function of surface roughness for

various F/D values. [30]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECltNIQUES 12
t

9

1

1(

9

I I I 1 1 I i I 11

I0

9

8

7

6

5

4

3

2

!

0

0.02 0.04 0.06 0.08 0.10

(a)

8 m

7-

6--

S--

4

2--

0

 soEco, 2_d •o98 _.

|jr 6

o s

_. PEAK 4

3

2

1 I

0 0

0 0.O2 0.O4 0.06 O.O8 0.10

4 r_,/_

(b)

Figure 2.1-3. Peak gain loss and side lobe level increase as a function of surface

errors (ET =-lOdB, F/D =0.7). (a) P= 1. The levels of the first,

second, and third side lobes for the undistorted case are -22.32, -29.35,

and -33.83 dB, respectively. (b) P = 2. The levels of the first, second, and

third side lobes for the undistorted case are -27.06, -30.83, and -33.85 dB,

respectively. [30]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 13
and peak loss to increase as t/,_ increases. An important design consideration from the

study is that as the no-error side lobe level decreases, its dependence on the surface e/,_

increases, requiring much smaller t/2 to maintain the low side lobes. We can also

extract a rule of thumb from these plots. The threshold for loss of about 0.5 dB or less

and a side lobe degradation of about l dB is an rms surface error of0.05,_ = 2/20. Larger

errors give rapidly decaying performance.

GENERAL ROUGH SURFACE EFFECTS ON THE RADIATION PATTERN

Shifrin presented some very detailed statistical analyses of random phase errors in

the aperture. [45] The aperture amplitude distribution is assumed to be uniform in the

absence of errors. The statistics of the errors are assumed to be Gaussian and

stationary. The mean and variances are constant across the aperture.

Shifrin showed that the presence of phase errors results in smoothing of the

radiation pattern. The nulls fill in and the main beam decreases. As the variance of the

phase errors increases, a progressive change in the radiation pattern occurs. The pattern

changes from an oscillating function to a monotonically decreasing function and the

power radiated in the principal direction decreases (the directive properties approach

that of an isotropic radiator [4.5]). As the correlation distance, c, of the phase errors

increases across the aperture, the radiation pattern approaches that of a system without

errors. [45] For small correlation distance c, the reduction in gain is caused by an

increase in side lobe level. [45] Finally, for large correlation distance (as with thermal

distortions or ribbed construction techniques), the reduction in gain is caused by an

increase in the beamwidth. [45]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 14
2.1.2 RADIAL SURFACE ERRORS

Deterministic surface errors are discussed in Sect's. 2.1.2, 2.1.3, and 2.1.4. The

discussion deals mainly with deterministic errors that are periodic.

Dragone and Hogg [9] presented an interesting theory on how phase errors affect

the radiation patterns of aperture antennas by using a Fourier series expansion of the

phase error. We will present a brief treatment of their approach and findings. Consider

a circular aperture shown in Fig. 2.1-4 with an aperture field distribution [9]

f(p) = [l - (2o/D)2Je (2.1-9)

where a is the amplitude of a square law taper, p is the radial variable, D is the diameter

of the aperture, and _(p) is the phase error (assumed to be a function of radius only).

The radiation pattern for this aperture distribution is [9]

g(u,¢)= 4 Jo (2.1-1o)

where u = (nD/,_) sin 0 and r = 2olD and _, is now expressed as a function of r. Upon

integrating with respect to $', the expression becomes [9]

g(L/) = --_ (l -- r200re'/_(r) Jo(IAr) dr

If O(r)= 0 the pattern for no phase error results and it is given by [9]

(2.1-11)

g(u) = 2(I - or) _ + 4_ u2 (2.1-12)

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 15
P

Figure 2.1-4. Circular aperture and the coordinate system. [9]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 16
Figure 2.1-5(a) shows this pattern for _--0.684 which corresponds to a -10 dB edge

taper.

For _(r)4= 0 , _b(r) could be expanded into a trigonometric Fourier series. First

consider a single term of a Fourier series [9]:

_b(r) = • m cos(2nmr) (2.1-13)

where O, is the magnitude of the sinusoidal phase error and m is the number of periods

along the aperture radius. The cases for m= 6 and rn= 12 with O, = 2n/16 were

computed and the radiation patterns are shown in Fig. 2.1-5(b),(c). The patterns show

that a sinusoidal phase error causes a large rise in the side lobe level only over a small

angular region. The angle where the region is located increases with an increasing

number of fluctuations and the amplitude of the rise is dependent on O,. [9] The effects

on gain loss as a function of the correlation Icngth of random surface errors (Sec. 2.1.1)

can be related to the period of the radial surface error. For the longer period radial

surface errors, the gain loss is caused mainly by increasing beamwidth. For the shorter

period radial surface errors, the gain loss is caused mainly by increased side lobe level.

Dragone and Hogg related the angular location of the disturbed region in the

radiation pattern to the number ofperiods ofthe phase error. The angular location of

the high side lobe region Oh is given by the grating formula [9]

2m

sin(0h)= +

(2.1-14)

where no = D/2.

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 17
o

v

; t

,.0

c=,

o

#

o

o

o

DEC_@EL$ DOWN FROM PEAK

!

ww

I

,_..__

/_:::r

0 _:_

0

o_

0

_'_

L.

oN

za

,_._._

=6

_ d

_.--

"_N

o_

,-. e_

0 _

_ 0

m_

4,)

L

,m

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES f8
2.1.3 AZIMUTHAL SURFACE ERRORS

Azimuthal surface errors arise in large space reflectors primarily from the structural

design. A radial rib reflector is an example of a design that gives rise to surface errors

that are periodic in an azimuthal direction. The azimuthal surface error for the radial

rib reflector gives rise to a periodic scalloped phase error. This scalloped phase error

function is sometimes described by the term "pillowing". Analysis for axisymmetric

reflectors is presented in this section. Offset radial rib reflectors have also been studied

and it was found that they exhibited the same behavior as axisymmetric radial rib

reflectors. [18, 281

Modeling the Radial Rib Reflector

An often used model for the axisymmetric radial rib reflector was presented by

Ingerson and Wong. [14] The ribs of the axisymmetric radial rib reflector lie on a parent

paraboloid of focal length f, The gores between the ribs are modeled as parabolic

cylinders of focal lengthf_. The f, andf_ are related by

(2.1-15)

where N_ is the number ofribs. The projection of the reflector and the geometry defining

a gore are shown in Fig. 2.1-6. The intersection of any plane containing the z-axis and

the gores on opposite sides of the reflector define a set of parabolas whose focal lengths,

f,, lie in a continuous distribution; and f, <f, <f,. [14] The optimum feed location in this

distribution was found to be [141

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 19
(a)

(b)

Z

¥

Figure 2.1-6. The model ['or an axisymmetric radial rib reflector. (a) The projected

aperture of the reflector. (b) The geometry of a gore. [14]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 20
A=f( I 2 2

3 ) (2.1-16)

wherefs is the distance the feed is from the vertex of the reflector. A later study showed

that this formula is good for reflectors up to 60 wavelengths in diameter and that a better

feed position can be found which optimizes the boresight gain for larger reflectors. [27]

The azimuthally periodic surface errors caused by the gore model of Ingerson and

Wong give the radiation pattern for an axisymmetric radial rib reflector some interesting

characteristics. Figure 2.1-7(a) shows the radiation pattern for two "equivalent"

reflectors. One is a 100_l diameter smooth paraboloid with F/D-0.4. The other is a

1002 wavelength diameter radial fib reflector with the parent paraboloid of the ribs

having F/D = 0.4. The feed pattern was chosen to be cos"0 such that the edge taper was

-10 dB. The two cuts for the radial rib reflector correspond to the center of the gore

(_b = 0 °) and a quarter of a way along the gore (#b = 5.625°). The main beam broadened

for the radial rib reflector. The effects of the surface errors on the side lobe structure

consisted of reducing up to the first eight side lobes and producing a large side lobe

envelope further out. The large envelope for far-out side lobes is due to the surface

errors exciting pattern expansion terms with Bessel functions J. of order n = N,, 2N,.

[27] Figure 2.1-7(b) shows a computed cross-polar radiation pattern. The effects of the

surface errors degrade the pattern somewhat, but the peak cross-pol level still depends

mainly on the feed cross-pol level. [27]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 21
O "x

\

/'

!

o

N it

I

° i

_r

!

0

01 •

I •

/

/

/.

6

ANGLE (degrees)

(a)

0

e_

0

0

t

@

ii

r!

ww

Nu

It

o i

%

x%

4 @ 8 10

ANGLE Cdegz-ees )

(b)

Figure2.1-7. Radiation patterns for 100 wavelength diameter reflectors with

F/D = 0.4 for a smooth paraboloid (solid line) and a ribbed reflector (16

gores) for _b= 0° (uniformly dashed line) and for _ = 5.62.5° (dot-dash

line). (a) Copolar pattern. (b) Cross-polar pattern. [27]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 22
PARAMETRIC STUD Y OF THE EFFECTS OF THE AZIMUTHAL

ERRORS CAUSED BY THE RADIAL RIB REFLECTOR

A parametric study was performed by Olver and Lizius to determine the influence

of the electrical and geometrical parameters of a radial rib reflector on the radiation

pattern. [27] They found that as the number of ribs increases (for a fixed diameter

reflector), the phase errors created by the gores decrease and the radiation pattern tends

toward that of a smooth reflector (the surface is approaching that of a smooth reflector).

For a fixed number of ribs, as the reflector diameter decreases, the efficiency tends

toward that of a smooth reflector, the beamwidth normalized to diameter becomes

narrower, the first few side lobes increase, and the gore related side lobe structure

decreases. The efficiency of radial rib reflectors was found to be relatively insensitive to

variation in the edge taper. This is due to the largest phase errors caused by the gore

surface occurring toward the edge where the feed illumination is the lowest and the

suppression of the first few side lobes which are normally affected by the edge

illumination. Greater distortion from the surface errors occur in reflectors with low

F/D ratios and as F/D increases, the pattern tends toward that of a smooth reflector.

It was also found that the scanning performance of a reflector using lateral feed

displacement is nearly identical to that of a smooth reflector with equivalent dimensions

(the analysis was for lateral displacements up to 10 wavelengths).

2.1.4 RADIAL AND AZIA'IUTHAL SURFACE ERRORS

A reflector structure that gives rise to both radial and azimuthal surface errors is

the cross-catenary umbrella reflector. [10] The gore surface is supported by a series of

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECtlNIQUES 23
the points anchored to cross-catenaries between ribs (see Fig. 2.1-8). The gore surface

consists of rectangular and trapezoidal warped, ruled facets instead of a triangular

section of a parabolic cylinder. The comers of the facets lie on the parent paraboloid

of the ribs. This antenna more closely approximates the parent paraboloid than does a

radial rib reflector. The tie-point related (azimuthal error) side lobes are generally less

significant than those from the radial rib reflectors. [10]

For a symmetrically illuminated cross-catenary umbrella reflector, approximate

analysis predicts the grating lobes caused by' the surface errors are located at [I0]

180m 

0¢_. nA e ° ,mc= I, 2, 3, ... (2.1-17)

for the catenary spacing, A, and [I0]

180mt}t

0 t,_ nA t ° , rnt = I, 2, 3, ... (2.1-18)

for the tie-point spacing, A,.

2.1.5 SUMMARY OF SURFACE ERROR EFFECTS

Section 2.1 reviewed works that deal with how surface errors affect the radiation

patterns of reflector antennas. The surface errors give rise to amplitude and phase errors

in the aperture plane. For relatively small surface errors the amplitude error effects are

minimal. The major effects are caused by the phase errors. Surface errors produce:

• Gain loss '

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 24
\

I

• T=e

X

Figure 2.1-8. Geometry of the cross-catenary umbrella reflector. [10]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECltNIQUES 25
• Null filling

• Increased side lobe levels

There are two traits that are common to the analyses of the different types of errors in

the previous four sections (Sec.'s 2.1.1, 2.1.2, 2.1.3, and 2.1.4):

• As the surface error amplitude increases, the gain loss and side lobes increase.

As the spatial period (correlation length) decreases, the angular location of the

region where high side lobes are produced moves farther away from the main beam.

2.2 FOCAL REGION FIELDS

Focal region fields are discussed in this section. Knowledge of the focal region

fields and the focal plane distribution (FPD) is essential in understanding the

compensation techniques reviewed in Section 2.3 and the scanning theory of Chapter 3.

(Note: The terms "focal plane distribution" and "focal plane fields" refer to the fields

produced in the plane containing the focal point of the reflector on reception. The term

"aperture plane fields" refers to the fields produced in the plane containing the focal

point on transmission.) The plots presented in this Section provide insight into how the

energy is distributed in the focal region due to rough surface distortion or scanning the

reflector. This insight is used to determine the size of the array feed and the element

spacing required to capture the scattered energy and coherently combine it to improve

the radiation characteristics of the antenna.

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 26
The analyses presented in this section assume that the focal plane fields are caused

by the physical optics currents induced on the reflectors by an incident plane wave. For

large reflectors (focal length and diameter much greater than a wavelength), the FPD

may be computed to very good approximation using the P.O. currents out to tens of

wavelengths from the focal point. [58] Farther out, the field contributions from

secondary currents (currents from one part of the reflector that are induced by radiation

from other parts of the reflector) and those arising from the finite size of the reflector

must be taken into account. [58] The compensation and scanning techniques discussed

in this study are not concerned with these focal plane fields far removed from the focus

and the physical optics approximation is considered to be valid.

2.2.1 FOCAL REGION ANALYSIS OF SMOOTH AND ROUGH

REFLECTORS

Geometrical optics analysis of a smooth parabolic reflector predicts that the rays

of a normally incident plane wave focus to a point, the focal point. This approach is

exact as the antenna becomes very large in terms of a wavelength and performs quite

well for optical devices. For radio frequency antennas, though, diffraction theory is

required for accurate evaluation of the distribution of energy in the focal plane. [31]

AXISYMMETRIC REFLECTORS

Rudge and Davies [31] presented the double Fourier transform compensation

technique for a cylindrical reflector (singly curved) with a rectangular aperture. The

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 27
coordinate system is shown in Fig. 2.2-1 (Note: The angle and distance variables are

defined in the Fig. 2.2-1. The circle in the figure is located in the aperture plane. The

circle does not indicate the perimeter of the reflector). In that paper, expressions for the

one-dimensional FPD derived using scalar diffraction theory were presented. Assuming

that the incident uniform amplitude, uniform phase plane wave was arriving along the

antenna boresigh't, the I-D FPD was found to be [31]

E(x)_2/_ sin kx_ ,,

kx/_ ' p <0.5 (2.2-1)

and [3 II

E(x) = _Jo(kx) , ,_ = 1.0 (2.2-2)

where _ is the maximum value of p = sin 0 cos _b which occurs at the reflector rim.

Figure 2.2-2 shows some plots of(2.2-1) and (2.2-2).

Rudge and Davies investigated the effects of periodically and randomly distributed

errors on the cylindrical reflector. The small reflector surface errors they considered were

evaluated as effective phase errors in the aperture distribution. Assuming a periodic

phase error of[31]

(2.2-3)

A

where _, is the maximum amplitude of the phase error and N is the number of periods

of the error across the aperture. The FPD was found to be [31]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 28
/

J

I

aperture plone

P

Figure 2.2-1. Coordinate system for the FPD analysis of Rudge and Davies. (Note:

the circle in this figure is located in the aperture plane. The circle does

not indicate the perimeter of the reflector). [31]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 29
IEI

O.1

IvJ

O-1

IEI

/

I

3or

i

2o,

1"0"

/

,/,

b

!

0 10

¢

Figure 2.2-2. Focal plane distributions for reflectors with

F/D = 0.5, and (c)F/D = 0.25. [311

(a) F/D= 1.0, (b)

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECtlNIQUES 30
and

E(x)= ,) sin s n ^-7-+ so(p,)

{ sin(ss ++ nNn) + sin(ss ---nNnnNn) }-]j < 0.5

(2.2-4)

A

E(x) = n Jo(_.)(s) + nJn(_e){Jo(s + nNn) + Jo(s- nNn)}

n,,,O

ix

, p-- 1.0 (2.2-5)

where s = k_x. The effect of the periodic phase error is to reduce the undistorted FPD

and redistribute the energy in a series of grating lobes. An example of an FPD with

periodic phase errors is shown in Fig. 2.2-3(a). Rudge and Davies also performed some

analysis with random surface errors. Figure 2.2-3(b) demonstrates the effect of random

phase errors by showing the mean FPD envelopes for a reflector with the surface rms

error shown versus an undistorted reflector FPD.

Rudge and Davies made some important conclusions from their analysis. For

reflectors with F/D > 0.5, there exists a direct Fourier transform relationship between the

aperture plane field on transmission and the focal plane distribution on reception. There

also exists a direct Fourier transform relationship between the aperture plane field and

the radiation pattern of the reflector on transmission. "The focal-plane field distribution

and that of the antenna radiation pattern are therefore identical, being related by a

double Fourier transformation." [31] This approximate relationship is exploited in

several compensation papers to be discussed in Section 2.3. Using this relationship and

knowing the effects of reflector surface errors on the radiation pattern (see Section 2.1.5)

Rudge and Davies were able to make the following conclusions [31]: Reflector errors

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 31
2

_Jo r/B)

N 0 N 2N

(a)

o,

w_.aO

-3K

1

o

t/x

(b)

Figure 2.2-3.(a) The focal plane distributionfor a reflectorwith a periodic phasc

crror. (b) Thc mcan focalplane distributioncnvclopcs fora rcflcctorwith

random surfaceerrors.[31]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 32
with long spatial periods cause errors in the radiation pattern near the main beam and,

hence, (using the above relationship) cause errors in the FPD near the focal point.

Errors with short spatial periods cause errors in the radiation pattern in regions far from

the main beam and, hence, cause errors in the FPD in regions removed from the focal

point.

Blank and Imbriale [4] presented a compensation and beamsteering technique for

circular paraboloidal reflectors. They included analysis of the FPD to provide insight

into the effects of surface distortions and beamsteering. The focal plane distribution

analysis is similar to Rudge and Davies [31] but they extended it to two dimensions. The

reflector geometry is shown in Fig. 2.2-4(a). For an incident plane wave, a smooth

parabolic reflector with FID > 1.0, the FPD is [4]

E(r, _') _. 2_(_) 2 Jl(k,_)

kr_ (2.2-6)

where _--sin _ and _ is defined in Fig. 2.2-4(a). The FPD for a smooth reflector is

plotted in Fig 2.2-4(b). For distorted reflector analysis, an approximation to a

gravity-induced distortion was chosen. The reflector is assumed to have a distortion

profile of [4]

A_ --e ,os cos(L_) (2.2-7)

where e is the amplitude of the surface error, s is a real number, p is the normalized

distance from the z-axis to a point on the reflector normalized to the reflector radius, L

is the integer number of periods in the distortion, and ( is the azimuth angle in the

aperture. The FPD for a distorted reflector is shown in Fig. 2.2-5. The spreading of the

energy away from the focal point is easily seen in these plots.

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 33
/

/,

v \

I

I

I

I

Zo

I I

I FOCAL I

iP PLANE I

\'\ i t

N \ , _ I=,.-FAR.FIELD \ 7,_ _,,o,_

, , ,X\ i / f,,i

\\i/\o ;xf

¢ /,/F " i/] v +

/ o_-f t

/ /

/ /

/

(a)

-I.;I0

-120

e,o°

F

I !

I I I ] I I I i

AGO 120

I,X

(b)

Figure 2.2-4. (a) The reflector geometry of Blank and Imbriale.

distribution for a smooth reflector with F/D = 1.0. [7]

(b) Focal plane

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 34
t _ 0.I_, s_ 1.0

120

0._

-1.20

-1.2Q

I I I I I I I I I

O.O0 1._

(a)

e - 0.2_, s-l.O

1 20 -t2_

.t.20 GOG 170

(b)

Figure 2.2-5.Focal planc distributionsfor a rough rcflcctorwith L = 3 and the e and

s indicated.[7]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECtlNIQUES 35
OFFSET REFLECTORS

Bem [5] analyzed the focal region field of an offset parabolic reflector. A very

important result from his study is that the focal plane fields of an offset reflector of

equivalent focal lengthf' (f' is the distance from the focal point to the reflector surface

along the z-axis) are nearly the same as the fields for an axisymmetric reflector with the

same focal lengthf=f' . [5]

2.2.2 FOCAL REGION ANALYSIS OF A SCANNED REFLECTOR

Scanning a reflector is implemented by moving the feed away from the focal point

or by weighting and possibly moving a phased array feed. The location of the feed or

the weighting of the elements of the phased array feed are found through some process

that involves a knowledge of the focal region fields when receiving a plane wave incident

from the desired scan angle.

SCANNED FOCAL REGION FIELDS, SMOOTH REFLECTOR

Rusch and Ludwig [32] studied the scanning of two paraboloidal antennas with

diameters of 34 and 68 wavelengths. The main purpose of the paper was to determine

the maximum scan-gain contours and to relate them to the Petzval surface. They

presented a plot of the amplitude and phase contours of the focal region fields in the

plane of scan (see Fig. 2.2-6) The equiphase contours are separated in phase by 180

electrical degrees. The plots are shown for the cases of reception of a plane wave on-axis

and for the angles of 8° and 16° to the right of the reflector axis. For on-axis reception,

2.0 REFLECTOR SURFACE ERRORS &ND COMPENSATION TECHNIQUES 36
x-_J_ 5CAN

_)lO• 34• O,433 WAVELE3qGTHS

EQUIP_ St

CONTOUR

WAVI[U_GTHS

TO LEFTOF --

FOCUS

I

/

p_rlZVAL_ _"

SURFACE

//

-3

WAVIB.ENGTHS

TOWARD

RffU[CTOR

IN,COMINGWAVE INCOMING WAV_ INCOMING WAVE |6 OFF-AXIS 8" OFF-AXIS ON-AXIS

-4,211dg -1.32dg

Figure 2.2-6. Amplitude and phase contours of the focal region fields in the

plane-of-scan. The scanned plots are for a plane wave incident from the

right of the reflector axis. [321

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 37
the maximum focal region field intensity is, as expected, at the focal point. As the scan

is increased to the right, the point of maximum field intensity moves to the left of the

focal point and eventually away from the reflector. The maximum field intensity

decreases as the scan angle is increased. The energy in the focal region is spreading out

with increasing scan angle.

Hung and Mittra also presented some focal region field contour plots in their paper

on wide-angle scanning. [12] They used the plots to determine the location and element

spacing of a feed array used to give improved scanning characteristics. The analysis was

performed on a 360 inch reflector with F/D = 0.48 at the frequencies f= 18 and 2 GHz.

The reflector was scanned at angles Osc,_ = 2.5 and 5 °.

The coordinate system used by Hung and Mittra is shown in Fig. 2.2-7. The

amplitude contour plots are shown in Figs. 2.2-8 and 2.2-9. Figure 2.2-8 shows the focal

region field distribution in the plane of scan while Fig. 2.2-9 shows the focal region fields

in planes that are parallel to the focal plane. As expected, the energy spreads out with

increasing scan.

SCANNED FOCAL REGION FIELDS, ROUGH REFLECTOR

Blank and Imbriale [4] further illustrated the dispersion of the energy in the focal

plane by plotting the FPD for a scanned, rough reflector. The surface error of (2.2-7)

was used with s = 1.0, L = 3, and a scan of approximately half of a beamwidth. The

FPD plots for their smooth and rough reflector are shown in Fig. 2.2-10. Comparing

to the plot for e = 0.12). from Fig. 2.2-5, the FPD is seen to be even more distorted than

that for a smooth reflector.

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 38
.................... • • • (laD,t4 r3x •

................

_y

Figure 2.2-7. The reflector geometry ['or Hung and Mittra's analysis. [12J

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 39
-?.It -?J --U --U

(a)

-tJI o

-IJI-

.._.

UO.S.

I_)'

LS

U

_LO

_._

.n_lO

, 40

4itO

i j,_ ._7_0___ _ . ,

' _ _ i 400

i_lzo i

_t20 _ i

ij _ ill

;;*o ' L '

_ o

q

i I

' _t20

/ -ee

** 1b.._ i , ,o_

, _; .-_,_. ._. ,_ !_

"*_--_.,;_--_- -,-. --- -;,, -., -_.- -,M -_ -,Lo-,,_ -,,_

X _ x_,t*,)

(b)

ORIGIP-JALP_GE IS

OF POOR QUALITY

Figure 2.2-8. Focal region distribution in the plane of scan for 18 GHz case. (a)

Incoming wave from 2.5 ° (24 beamwidths). (b) Incoming wave from 5.0 °

(48 beamwidths). [12]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 40
U •

I

It@

all.0

(a)

L

!

" - L

" "_ (_° .._._-_.__/..'/.,,'

,-Oili.-¢Oi5 _ _ --liO "-i8 -41',0 -MIJi -sli_ -IJ5 -iJi -ICJi -ICR

(b)

"._,;,h',,_, _2 _'_'_r "_ E,¢,._

O/_ #O_,E_' "i-:- _,.-.,_LiT"lf

Figure 2.2-9. Focal region distribution in planes parallel to the focal plane for the

18 GHz case. (a) Incoming wave from 2.5 °, z=0.15 inches. (b) Incoming

wave from 5.0", z=0.6 inches. [121

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 41
I'Oi" I1/ _, :14_

- -t?

_- o.0

lU.

11 I L

°1'_.1,0 _Q 10

(a)

o

.1.o o.o o

(b)

Figure 2.2-10. Focal plane distributions for scanned (a) smooth and (b) rough

reflectors. The distortion profile is that of (2.2-7) with s = I and L -- 3.

[7]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECtlNIQUES 42
2.3 PAST ELECTROMAGNETIC COMPENSATION

TECHNIQUES

In this section available electromagnetic compensation methods are reviewed.

Computational and experimental techniques are presented. All the computational

techniques require accurate knowledge of the reflector surface while the experimental

techniques do not.

2.3.1 DOUBLE FOURIER TRANSFORM TECHNIQUE

Rudge and Davies wrote one of the earliest and most referenced papers on

electromagnetic compensation for surface errors. [31] It followed a work by Davies [11]

where techniques for electronic compensation of surface errors were proposed. Rudge

and Davies' technique exploits the Fourier transform relationships between the aperture

field distribution and the radiation pattern on transmission and the aperture field

distribution and the focal plane fields on reception. They showed that the function for

the focal plane fields on reception is very similar to the function for the far-field

radiation pattern for reflectors fed from the focus with F[D > 0.5 [31]

Rudge and Davies" compensation solution requires taking a second spatial Fourier

transform (the first being the Fourier transform of the received aperture field by the

main reflector to get the focal plane fields) on the signals received by a multiple feed

array (see Fig. 2.3-1). The signals produced at the output of the Fourier transforming

device correspond directly to the aperture field distribution. [31] If a plane wave is

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 43
flector

___ plane On corn_ned

_l__ I_l$;gool- 1°_tput

_Z]tronsform 1_6_jcombiningjl_ --I_-I matrix

_ network /

I n-element phase

jarray shifters

Figure 2.3-1. Feed system for implementation of Rudge and Davies' surface error

compensation technique. [31]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 44
incident from boresight, the focal plane distribution for a smooth parabolic cylinder at

the input of the Fourier transforming device looks something like a "sin x/x" function.

For a smooth reflector, the output signals of the transforming device are uniform in

amplitude and phase. For a distorted reflector, the signals are nearly uniform in

amplitude and the deviation from constant phase is due to the surface errors.

Compensating for the surface errors is performed by measuring the phase at the output

of the Fourier transforming network and coherently combining the signals using phase

shifters (O_- O4 in Fig. 2.3-1). [31]

The initial implementation of Rudge and Davies' compensation method was

restricted to one dimension. This allowed the compensation technique to be

experimentally implemented via a Butler matrix. The Butler matrix is a device that

consists of a matrix of fixed phase shifters and couplers (which are lossless in ideal form)

whose output is a sampled spatial Fourier transform of the field distribution applied

across the input. [31] Fig. 2.3-2(a) shows an example of the sampled Fourier transform

of a hypothetical focal plane distribution. Practical implementation of this feed system

requires using a feed array that is small enough to prevent blockage problems. Therefore

the only kinds of surface errors this method (and all methods) can compensate for are

ones which have slowly changing variations across the reflector surface (giving rise to

focal plane errors near the focus, see Sec. 2.2.1). Even though the focal plane field is

sampled with a finite aperture, it gives a good approximation to the infinite Fourier

transform when the focal plane distribution decay is steep enough. [31]

The Butler matrix feed system with adjustable phase shifters was used in an

experiment with a variable-profile parabolic cylinder reflector with F/D = 0.5. A second

Butler matrix was used as the combining network. The output signal was taken off the

port of the second Butler matrix that had maximum output (see Fig. 2.3-2(b) for a

schematic of the feed network). The optimum spacing, 60, of the elements is [31]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 45
I

X

TTTT1']'tT

Butler _rix

P

(a)

ir_ut

ports

I

Drimary-feed orroy

IIITTTT 

Dhl3Se shifters

outout

looOs

(b)

Figure 2.3-2. (a) Sampled Fourier transform implemented by a Butler matrix. IEI

represents the value of the focal plane field at the input of the matrix.

I FJ represents the corresponding output of the matrix. (b) Schematic

diagram for Rudge and Davies" experimental 8-element array feed system.

The top Butler matrix takes the Fourier transform of the focal plane

distribution. The phase shifters are used to coherently phase the output

signals of the matrix. The second Butler matrix is used to recombine the

coherent signals. [31]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECIINIQUES 46
a0= (2.3-1)

where t) = sin 0 and 0 is the angle between the parabola axis the ray from the focus to

the reflector edge. This is the maximum separation of the feed array elements which

prevents grating lobes from occurring in the angular region subtended by the reflector.

[31] The variable-profile reflector built by Rudge and Davies is capable of imposing a

phase error, fl,, in the aperture of [31]

P

(2.3-2)

^

where p, is the amplitude of the phase error, N is the number of periods of the phase

error across the aperture, p -- sin 0 cos _b (see Fig. 2.2-1), and ,_ is the maximum value

ofp at the reflector rim.

The results of experiments using the feed system to compensate for phase errors

are shown in Fig. 2.3-3 for the three cases of a smooth and two distorted cylindrical

reflectors. The reflectors had F/D = 0.5 , D = 180cm at I0 GHz. Compensation was

performed with a Butler matrix with 8 elements. The close Fourier relationship between

the focal plane field and the radiation pattern is illustrated in Figs. 2.3-3 (a) and (b) by

comparing the magnitude plots. The double Fourier transform method is shown to be

very effective in compensating for the surface errors by comparing the Figs. 2.3-3(c), (ii)

and (iii), to that of the smooth reflector, Fig. 2.3-3(a), (i).

The authors presented a rule of thumb for determining the compensating range

of their adaptive feed. For aperture phase errors with 0 < ,6, _ 7t/2 and with spatial

frequency less than (n- 1)/2, the n-element array feed is able to compensate for errors

with [31]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 47
1.OI

v

w

0-5

O

1.O

0.5

I

-2_ 0 2. 0

I

Q

0

0

V_

1

-2" f 2 °

e _deg

b

f

J (_}

0

F

-2" 6 2"

I

e, deg

c

(ii)

(i_i)

Figure 2.3-3. Experimental results for the variable profile reflector using the feed

system of Fig. 2.3-2b. The three cases are for (a) focal plane field, (b)

radiation pattern for single feed, and (c) radiation pattern with array feed

adjusted for surface error compensation involving (i) no profile errors, (ii)

A

phase errors with p,=0.5rr and N= 2, and (iii) phase errors with

/_, = 0.4n and N = 3. [311

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 48
_eN < n - 1 26_

_ (---j--) (2.3-3)

where n is the number of phase-compensated inputs to the summing network and 8 is

the element spacing.

In their conclusions, Rudge and Davies cited practical problems for implementing

their technique in two dimensions. Butler matrices would probably be too hard to build,

too lossy, and too expensive for two dimensional problems. They recommended seeking

other alternatives to perform the Fourier transform, such as microwave lens or

multi-reflector techniques. [31]

2.3.2 CONJUGATE FIELD MATCH TECHNIQUE

(COMPUTATIONAL)

Conjugate field match (CFM) for reflectors with array feeds is the most widely

published of the surface error compensation techniques. CFM has also been extensively

used in reflector scanning analysis. The method is based on maximizing the power

received from a given direction. The drawback to the computational CFM technique is

that you need an accurate model of the reflector surface.

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 49
CONJUGATE FIELD MATCH BACKGROUND

Much of the published work on conjugate field match deals with primary feed horn

design. This subsection presents a background summary of the CFM method applied

to horn design in preparation for application to reflectors with array feeds.

The CFM formulation was, in part, derived from a study by Midgley [25] dealing

with reradiation of a rectangular pyramidal horn. In that study, the conjugate field

theorem [25] is used to define complementary waves for which a given antenna receives

all the incident power. [25] "l fan ordinary aerial emits a certain outward travelling wave

f(jkr-jcot), it is at least possible to imagine a structure capable of returning a

complementary wave f(jkr +jwt) of equal strength, in response to which the original

aerial is a perfect absorber." [25] For example, consider a reflector feed. If the

transmitting field distribution in the feed's aperture is the conjugate of the fields when

receiving and either the E-field or H-field is reversed, the feed will absorb all the energy

incident upon the aperture. [26] Perfect absorption would correspond to an antenna

with an aperture efficiency of 100%. The goal is then to closely match the field

transmitted by an antenna to its complementary received field.

A measure of the match of the actual transmitted field of an antenna to its

complementary field may be represented by the correlation of the two fields. The

correlation of the actual transmitted field and the desired complementary field can be

evaluated over a surface enclosing the receiving antenna. [25] The result is an integral

expression for the power received by an antenna which, upon normalization, gives the

aperture efficiency. [41] The integral expression for the power transmitted to the

receiving antenna, P_, can be derived in two ways. Both solutions are presented here.

First, consider two arbitrary antennas A and B. If A transmits power Pr and B is

terminated in a matched load, the power received by antenna B is [25]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 50
Px = aPt (2.3-4)

where 0c is a constant, a _ 1. It is a consequence of the Lorentz reciprocity theorem and

the Friis transmission formula that if B transmits Pr and A is terminated in a matched

load, the received power is again P_ in (2.3-4). [251

Let E,_, /t,, be the fields transmitted by A where the transmit power is Pr. Let

Ea, /_a be the fields transmitted by B where Midgley now assumes the transmit power

is 0_Pr. [25] The power received by A is now [251

Pte= a2Pr (2.3-5)

Using (2.3-4) and (2.3-5), [25]

=

s s

where the surface S encloses A, the integral enclosed by the left set of brackets represents

the power transmitted by A through S, and the integral enclosed by the right set of

brackets represents the power transmitted by B passing through S. The desired integral

expression is found by evaluating the elemental power passing through S. The elemental

expression is [25]

(ap_)2 ._[ ×HA.a - _][_,×_s.a - _] (2.3-7)

where 6P_ is the power crossing an element 6S enclosing A.

[251

Using the vector identity

(_×_).(_×_)--(_._)(_._)- (_._)(_._) (2.3-8)

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 51
in (2.3-7) and factoring gives [25]

(_,>'_- [(_ x,_).(_ x,_)](_. _)

-[(_,,×_,,,)x,5_].[(L x,_,,)x,5_]

(2.3-9)

The second term in (2.3-9) is zero if,_g is perpendicular to E_ x H, or E_ x/¢B. [251 This

condition is satisfied if the surface S is in the far field of A or B. Applying (2.3-8) to the

(nonzero) first term in (2.3-9) gives [251

(_p,)'=[(_. _:_)(_._)- (_:_._)(_. _)]_s_ (2.3-10)

For linearly polarized antennas with the far field E-fields perpendicular to the H-fields,

the second term in (2.3-10) is zero. If the magnitudes [25]

G D

H_ Ha -_/ (2.3-11)

where )7 is the intrinsic impedance of the medium, then (2.3-10) may be written as

2

_7

(2.3-12)

which leads to the desired integral expression for the power received by A [25]

s

dS (2.3-13)

The aperture efficiency of an antenna can be found by normalizing the fields in (2.3-13).

The _/ in the denominator of (2.3-13) is normalized by multiplying the equation by r/.

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 52
The transmit field EA is normalized to have unity power. The received field /_s is

normalized to have unity power intercepted by the aperture of antenna A. [41]

The second solution for finding the power transmitted to the receiving antenna has

been used in designing feeds and subreflectors for reflector antennas. [41, 56, 57] The

transmission efficiency T for prime focus reflectors is defined as the ratio of the power

delivered to a matched load connected to a primary feed to the total power intercepted

by the reflector aperture. [41] The efficiency T is found by applying Robieux's theorem

of the coupling of two antennas. [60] Wood [60] applied Robieux's theorem to focal

plane calculations giving [60]

1

FP

where b2/, /1, are the fields in the focal plane that would be present if the feed were

transmitting and the /_p, /tee are the received focal plane fields for a plane wave

illuminating, the reflector. The fields incident upon the focal plane are assumed to be

normalized to unity power. [60] If the E and H of the E,H pairs are perpendicular,

(2.3-14) reduces to (similar to [41])

,--

FP

where K is a normalization constant. Equation (2.3-15) is the second solution for the

P_ and is equivalent to (2.3-13) (when normalized) for fields evaluated in the focal plane.

The expressions in (2.3-14) and (2.3-15) have a unity maximum for the conditions

[601

_'f--- _'_t, (2.3-16a)

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 53
Hf= - H_p (2.3-16b)

Using (2.3-16) in (2.3-14) gives two expressions for the power density integration of the

received plane wave or the transmitted wave in the focal plane. The powers are, by

definition, normalized to unity and their sum multiplied by the 1/2 gives a maximum of

T-- 1. This implies that the aperture efficiency is maximum for the hypothetical feed

that presents a conjugate match to the received focal plane fields over the entire focal

plane. [60]

Of course real feeds are of finite size. The surface of integration is then assumed

to be the aperture plane of the primary feed instead of the entire focal plane. [41] It is

still assumed that the optimum aperture efficiency is realized over the aperture of the

feed if the conjugate match conditions of(2.3-16) are met. [60] ttowever, this assumption

has not been proved for the limited aperture case. [42]

The array feed implementation of the CFM method is an extension of the primary

feed matching. In the array-fed case, the focal plane fields are conjugate matched at a

set of discrete locations corresponding to the feed element locations. This

implementation is useful for matching the focal plane fields of scanned or distorted

reflectors (Sees. 2.3.2 and 3.2). The focal plane fields of scanned or distorted reflectors

are spread over a wider region than for unscanned smooth reflectors (see Sec. 2.2). The

array feed configuration allows for matching the focal plane fields over a broader area

and, hence, providing a higher aperture efficiency.

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 54
CFM COMPENSATION

The CFM compensation technique is implemented with a phased array feed. The

distorted reflector causes errors in the focal plane fields and the energy is dispersed in the

focal plane (see Sec. 2.2.1). The feed array is used to capture the distributed energy and

coherently combine it. The individual element excitations, A,, are set to be proportional

to the complex conjugates of the focal plane fields that would be received by the ith feed

element given a plane wave incident from the desired direction (boresight if it is not

scanned). [4]

There are two ways to determine the required CFM excitations A,. The A, may

be found by direct computation of the received focal plane fields (In the subsection after

the next subsection we call that Direct CFM or DCFM) or A, may be found by using a

reflector far field computer code (In the next subsection we call that Indirect CFM or

ICFM). [331

ICFM AND MODIFIED ICFM WITH IMPROVED SIDE LOBE

PERFORMANCE

Reciprocity can be used to show that the ith CFM feed element excitation upon

reception of a plane wave is proportional to the secondary field transmitted in the

desired direction from a reflector illuminated by the ith feed. [17, 4, 33] Rather than

directly computing the focal plane fields to determine the coefficients, the element

excitation coefficients are computed indirectly using a far field computer program and

invoking reciprocity. This is the basis of the indirect CFM technique (ICFM [2]).

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 55
Blank and Imbriale studied array feeds and the use of ICFM to compensate for

reflector surface errors. The authors derived some useful design formulas for realistic.

feed array design. The formulas were based on approximate circularly symmetric,

unidirectional feed radiation patterns of the form [4]

f(_b) = cosq'(_) (2.3-17)

The exponent can be related to the edge taper t of a single feed at the focus (0 < t < 1).

The exponent, q_, is approximately [4]

-8 In(t) (F/D)2 _ 0.5 (2.3-18) qt ,r, b

where b is a constant that depends on the effective aperture of the feed element. [4] This

value of qj leads to a minimum element diameter, dot, for a single feed of [4]

de, j.-8 In(t)

b (F/D) (2.3-19)

for a triangular grid array of circular elements (see Fig. 2.3-4).

The surface errors used by Blank and Imbriale were assumed to have distortion

profile Az, of (2.2-7). This error is based on data for gravity-induced distortions typical

of large ground based reflectors. [4] Using this surface distortion profile and the ICFM

technique outlined above, the authors looked at the ability to compensate for gain loss

for a variety of cases for various combinations of the number of feed elements, N; F/D

ratios; distortion profile parameters _ and s (e and s are empirically determined constants

for defining the gravity-induced distortion profile); and feed element parameter b. An

edge taper of-10 dB was used in this parametric analysis. The gain loss is referenced to

the maximum gain for a circular aperture (49.94 dB for D = 1002 ). Blank and Imbriale

found some interesting results during the study. For a single feed, the smaller the values

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 56
0.0

Xo),

Figure 2.3-4. An example of a 37-element triangular grid array. [7l

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES

57
of F/D, the smaller were the effects of the distortion whereas for the array feed, the

distortion effects are almost independent of F/D for the simulated gravity induced

distortion. [7] Gain loss increased with increasing surface distortion amplitude. [7] Also,

gain loss is less for errors concentrated at the reflector edge than for more uniformly

distributed errors. [4]

Blank and Imbriale used the results of the parametric analysis to derive expressions

for the optimum feed element diameters. For N = 7 [4]

z. a,l = D) (2.3-20)

and for N-- 19,37 [4]

d"'lgav~T =T (F/D) (2.3-21)

The feed array configurations using (2.3-20) and (2.3-21) are shown in Fig. 2.3-5.

To illustrate the compensating effects of CFM on a radiation pattern, Blank and

Imbriale used a 37-element feed array. The compensated pattern is compared to the

badly distorted pattern when a single feed is used (see Fig. 2.3-6).

Rahmat-Samii [33] worked with a modified ICFM algorithm that reduces the side

lobe level. Rahmat-Samii's ICFM uses a fictitious feed illumination taper when

computing the element excitations. This numerically produces an amplitude taper in the

aperture. The resulting secondary fields determined from the set of element excitations

lead to an array that compensates for the surface distortions while maintaining a

desirable side lobe level. The actual feeds will then be used to compute the far field

patterns. [33]

Rahmat-Samii presented a numerical example to demonstrate the compensation

algorithm and its effectiveness. A distorted 20-meter deployable offset reflector

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES $8
N=!

N=7

N= 19

= 37

Figure 2.3-5. Optimum feed array configurations for N = 1, 7, 19, and 37 elements.

[71

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECltNIQUES 59
!

I

I I I

- 37

\

\

\

\

\

\

Figure 2.3-6._ RadiatiOns patterns for a distorted reflector (F/D= 1.0, e = 0.12,_

-2.0). The N= l plot corresponds to the uncompensated far field

pattern using a single feed at the focus (d. = 2.1). The N= 37 plot is the

compensated pattern using the triangular grid array and CFM

(d. = 0.71). [7]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 60
operating at L-band had its edge displaced by 60 mm due to a simulated thermal

distortion. A 19-element feed array was used (see Fig. 2.3-7). The uncompensated

patterns are shown in Fig. 2.3-8 along with patterns compensated for gain only (similar

to Blank and Imbriale's approach [4], ICFM without side lobe control) and ones

compensated for gain and side lobe level.

Rahmat Samii also has worked with an experimental CFM technique. This is

discussed in Sec. 2.3.4.

DCFM WITH IMPROVED SIDE LOBE PERFORMANCE

Acosta [1] outlined a modified CFM algorithm which has improved side lobe

performance. The method uses the received focal plane fields to compute the excitations

and is called direct CFM (DCFM). To improve the side lobe levels, Acosta's DCFM

algorithm assumes that the incoming plane wave used to compute the focal plane fields

is fictitiously amplitude tapered. [11

Acosta, Zaman, Bobinsky, Cherrette, and Lee [21 compared this modified technique

to regular CFM results. The modified CFM coefficients were directly determined by

computation of the received focal plane fields assuming the amplitude tapered plane

wave is incident upon the reflector. The lower side lobe characteristics of the DCFM

were illustrated by numerical example. The reflector antenna with the distortion profile

shown in Fig. 2.3-9 was analyzed. Compensation was implemented with the 37-element

feed array of Fig. 2.3-9(b). The elements were positioned so that most of the power

spread over the focal region by the surface distortion was captured by the array. The

plots for the undistorted reflector (single feed), distorted reflector (single feed), and

compensated patterns are shown in Fig. 2.3-10. The plots in Fig. 2.3-10 show the

improved side lobe structure produced by the modified approach using the amplitude

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 61
2

• •

• •

• •

0

0

Figure 2.3-7. Reflector geometry with the simulated thermal distortion

19-element feed array. [33]

and the

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 62
50

4O

8

t&l

D

I--

--I

al.

:[

,<

30

2O

10

0

-10

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

e (d_i!

5O

¢-90 ° /_ ------ BEFORE

40 COMPENSATION

........

COMPENSATION

-J ,

30

/[i -I \

2

o L.I_'_i : "

3 20 "_" r4;..

o _._L_fi.. °/" a,a,._,

-6 -5 -4 -3 -2 °1 0 1 2 3 4 5 6

# Id_)

Figure 2.3-8. Radiation patterns for a distorted reflector whhout compensation

(dashed), with compensation for gain only (dotted), and with

compensation for gain and side lobes (solid), [33]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 63
s;o( view

'/I _RCFLECTOR

II Oft /

T_

4.7fL: FOCAL

_NT

(NO VI(W

tY

! @

l_vf

!

I

I

i

SNctng 1.1,_

- 37 [l_s

If

(a) (b)

-..;_:?_..L pAGE IS

01: poOR QUALITY

(c)

Figure 2.3-9. (a) Reflector geometry for the DCFM analysis. [8] (b) 37-element

customized feed array for the DCFM analysis, [2] (c) Distortion profile

for the DCFM analysis (height dimension in wavelengths, base dimension

in inches, f-- 30GHz). [81

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 64
44

0

-3 -2 -I O 1 2 3

(a) (b)

M

0. 3 -2 -1 0 1 | 3

O0

4O

N

(c) (d)

Figure 2.3-10. Radiation patterns for (a) undistorted reflector, single feed; (b)

distorted reflector, single feed; (c) distorted reflector, array feed, regular

CFM; (d) distorted reflector, array feed, Acosta's DCFM. [2]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 65
tapered plane wave to compute the excitations. Note, however, that the directive gain

is about 2 dB lower than the regular CFM approach. [2]

2.3.3 LEAST SQUARED ERROR TECHNIQUE

Bailey [6] developed a method for computing the excitation coefficients of an array

feed that give an approximation to a desired aperture field distribution. The author

notes that the method could also be implemented by computing the excitation

coefficients that approximate a desired far field pattern. The algorithm matches the

actual aperture field to the desired aperture field at discrete points. The method of least

squares is then used to compute the excitations. [6] As with computational CFM (See.

2.3.2), an accurate model of the reflector surface is required.

Aperture field information is used to compute the excitations for the least squared

error compensation technique. The aperture fields due to each of" the N array elements

when transmitting are computed. The desired complex aperture field, F(x, y), is

approximated in the xy-plane at I discrete points by [6]

At

F(xl, Yl) _" E A,_fn(xt, Yl) , i= 1,2,3, ... ,I (2.3-22)

,)1=,I

where N is the number of feed elements, A. is the nth element excitation, andf.(x,,y,) is

the field due to the nth element at location (x,,y,) in the aperture plane. The number of

field points in the aperture I must be sufficiently large to give a representative sample

of the aperture distribution. The array may be of arbitrary geometry but for this study

planar triangular grid arrays were used. [6] The complex excitations, A., are found such

that F(x, y) is approximated in a least-squared sense by the N-element feed array. [6]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 66
The quantity G is a measure of the error between F(x, y)

(2.3-22) and is defined as [6]

and the approximation in

Minimizing G with respect to A,, for m = 1,2,3, ... ,N gives [6]

OG -2 x_,y_ x_,y_) A,,f,(x_,y_ 0

OAm

(2.3-23)

(2.3-24)

which gives N equations with N unknowns. Defining [6]

I

= f,.(x, y )F(x,yt)

I=I

(2.3-2s)

=

t= l

(2.3-26)

and using (2.3-25) and (2.3-26) with (2.3-24), the problem can be expressed as [6]

nl

B_,

SI!

S21

Sm S m ... aNN

AI

A2

A^

(2.3-27)

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 67
where the Unknowns in (2.3-27), A,, may now be solved for the feed element excitations.

Bailey notes that this technique does not maximize the gain nor does it minimize the side

lobes but it approximates the desired aperture field distribution and therefore the desired

radiation pattern in all angular space within the pattern correction limits of the finite

N-element array feed. [6]

Bailey performed a computer simulation to illustrate the compensating capabilities

of this technique. He analyzed a single quadrant of the IS-meter hoop/column mesh

deployable antenna assuming a frequency of 6.4 GHz and 3.5 inch element spacings.t6]

Analysis for smooth, distorted, and compensated reflectors was presented for reflectors

with N= 7, 19, and 37-element triangular grid feed arrays. The actual surface

distortion of the 15-meter hoop/column reflector was measured and the distortion profile

is shown in Fig. 2.3-I I.

The geometry and initial excitations for a 7-element array fccd are shown in Fig.

2.3-12. The radiation pattern for the reflector distortion of Fig. 2.3-I l for the 7-element

array-fed case of Fig. 2.3-12 is shown in Fig. 2.3-13. Compensation analysis for the

7-element case showed negligible improvement. [6] The array feed geometry and

compensation excitations computed using the method of least squares of(2.3-27) for a

19-element feed are shown in Fig. 2.3-14. The improved radiation pattern for the

19-element compensation case is shown in Fig. 2.3-15. Further compensation analysis

was performed with a 37-element feed. The geometry and compensation excitations for

the 37-element feed are shown in Fig. 2.3-16. The much improved radiation pattern for

the 7-element compensation case is shown in Fig. 2.3-17.

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 68
Figure 2.3-11. Exaggerated distortion profile for the single quadrant of the 15-meter

hoop/column antenna. The rms of the surface distortion is 0.061 inch. [6]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 69
(deg)

Figure 2.3-12. Geometry and excitation coefficients for the 7-element array. [6]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 7O
2.0

1.fl

0.0

-1.0

-Z. 0

m.O a.O gO.O 91.0

e (degrees)

Figure 2.3-13. Uncompensated radiation pattern for the distorted reflector with the

distortion profile of Fig. 2.3-11 and the 7-element array feed excited as

shown in Fig. 2.3-12. [61

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECltNIQUES

71
Figure2.3-14.Geometry and compensation excitation coefficients for the 19-element

feed. [6]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 72
2.0

1.0

0.0

"O-

-1.0

i @

m

-2.0 . L , l , I i • i • I . , L * I _ i , . _

m.O 89.0 90.0 9L.O 9Z.0

8 (degrees)

Figure 2.3-15. Compensated radiation pattern for the distorted reflector with the

19-element feed. [6]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 73
Figure 2.3-16. Geometry and compensation excitation coefficients for the 37-element

feed. [6]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 74
2.0

0.0 -

-1.0

m

D

-Z.O

M.O M.O gO.O I1.0 _.0

Figure 2.3-17. Compensated radiation pattern for the distorted reflector with the

37-element teed. [6]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECiINIQUES

75
2.3.4 CONJUGATE FIELD MATCH TECHNIQUE

(EXPERIMENTAL)

Rahmat-Samii [35] performed an experimental implementation of the DCFM

compensation method. The experiment follows the same basic theory of the

computational DCFM technique (see Sec. 2.3.2). In the experiment, the received focal

plane fields for a distorted reflector were measured (instead of computed) and the

element compensation excitations were determined using CFM. The advantage of the

experimental technique is that the reflector surface shape need not be known.

The experimental CFM study was performed on a reflector that was shaped with

that of a simulated thermal distortion. The simulated distorted surface deviates from a

smooth parabolic reflector by the [35]

j_=O.Oll( 2p )3 --if- cos 2_ meter (2.3-28)

which describes the dominant term in a typical thermal distortion. The reflector

geometry is shown in Fig. 2.3-18. The feed array consisted of 16 cigar elements in a grid

spaced 1.06Jl apart at 8.45 GHz. [351 There was an analog phase shifter and variable

attenuator behind each element. [35] Considerable effort was spent generating

calibration curves for each element so that the desired amplitude and phase excitation

for compensation could be implemented despite intercircuit amplitude and phase

imbalance. 143]

The algorithm used to implement the experimental CFM approach is shown in Fig.

2.3-19. The experiment showed that the compensated distorted reflector gain improves

by 2.5 dB over the uncompensated case. [35] Figure 2.3-20 compares the patterns for

the uncompensated and compensated cases.

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 76
D- L_ m

REFLECTOR

SURFACE

I

I

[m I

I

I

,/

18.900 _

/

/

Figure 2.3-18. Reflector geometry and the simulated thermal

in the experimental CFM st distortion proHle Used

elements in a gr/d spaced I 06_ dy' The feed array consisted of 16 cigar

• apart at 8.45 GHz. [36J

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES

77
NE WAVE

SMART [

APPARATUS

THERE MAY NOT BE ANY

NEED TO KNOW WHAT THE

ACTUAL REFLECTOR

oIsTo_T_O N IS.

e MEASURE ELEMENT THE RECEIVED AMPLITUDE & PHASE OF EACH ARRAY

[ • CONJUGATE MATCH FOR THE ARRAY EXCITATION COEFFICIENTS[

. REceiveORRAO,ATEW,THENT,REAR.AY

Figure 2.3-19. Algorithm for implementing the experimental CFM technique. [361

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 78
0 = 00 _._ ... MEASURED

/_'f._ ' _ _'_ORIGINAL

"1 '%'_. L

,_,_>-25 / / J .,/ E 4

-35 ._ "_ A / .V

5 4 3 2 1 0 1 2 3 4

0 (dog)

0

LU

c_

.,,,,I

CL. I

,1I

LM

>m

t-.

<(

.,J

LU

it.

0 _ ,

-5 - 0 = 45° I

- 10 /_---'-_\/_ • . MEASURED

-15 - / / ' \\.ORIGINAL -

-2o - I \ -

-25

-30

-35

-401

5 4 3 2 1 0 1 2 3 4 5

(deg)

0

U4

0 -10

_- -15

,,.I

lb.

• -20

,(

-25

_- -3o <[

,,.I

ua -35

ee

-40,

5

0 = 90 °

\

\ MEASURED

/" , _ \ ORIGINAL

_.- ------/ , \ \.___

--_, \/COMPENSATED\/ X /_

1 1 1 i _ ', 1 i

4 3 2 1 0 1 2 3 4 S

Figure 2.3-20. Measured uncompensated and compensated radiation patterns. The

plots are all normalized to -5 dB but the gain of the compensated pattern

is actually 2.5 dB higher. [36]

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 79
2.3.5 SUMMARY OF COMPENSATION TECHNIQUES

The existing electromagnetic compensation techniques were discussed in this

Section. A summary of the techniques is presented in Table 2.3-1. The methods try to

correct for errors in the focal plane fields or the aperture distribution and, hence, correct

for irregularities in the overall radiation pattern. The compensation techniques allow for

some control over the side lobe envelope. They are restricted to compensating for

surface errors with long correlation lengths that cause errors in the focal plane fields near

the focal point or errors in the radiation pattern near the main beam. The pattern of a

rough surface reflector (very short correlation lengths) cannot be improved using

electromagnetic compensation.

Rahmat-Samii looked at the effects of random errors in the amplitude and phase

of the feed excitations for CFM compensation. [34] He found that with errors of ±0.5

dB in amplitude and +10 ° in phase, there were negligible effects on the compensated

radiation patterns.

Rahmat-Samii also noted that better improvement can be obtained using larger

arrays. [33] This statement is supported by Bailey's results in Sec. 2.3.3. The pattern for

the reflector with the 37-element array feed (Fig. 2.3-17) shows improved pattern

performance over that of the 19-element array feed case (Fig. 2.3-15).

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECHNIQUES 80
l; _4 r_ r_ r_

_J

.o"

_J

e.;

o

a_

g5

t,)

r_

,... o

• t,J

.'5

0

CJ

CJ

..c:

C....

0

C/_ 0

"0

0

w

"{:::

0

_d

¢.)

o_

e_

E'o

_ _o o

F"

._ _ ._ ._.

2.0 REFLECTOR SURFACE ERRORS AND COMPENSATION TECItNIQUES

81
3.0 REFLECTOR SCANNING

The ability to scan a reflector pattern is essential to the pattern synthesis

compensation technique (see Ch. 4). There are many methods used to scan reflectors.

Displaced feed scanning involves moving the feed from the focal point. Electronic

scanning uses a fixed or movable array of feed elements.

This chapter is a literature review of various reflector scanning techniques in

preparation for discussion of the pattern synthesis compensation technique, Sec. 4.2.

The different methods for scanning a parabolic reflector are outlined. The characteristics

of scanned patterns are discussed. The majority of this chapter is devoted to single

parabolic reflector antennas but the last section deals with dual reflectors.

3.1 DISPLACED FEED SCANNING

The main beam of a parabolic reflector can be scanned by displacing the feed from

the focus. The pattern degrades with increasing scan due to phase errors introduced by

3.0 REFLECTOR SCANNING 82
the displacement of the feed. These phase errors limit the range over which the reflector

can be scanned.

Approximate scalar analysis for planar and linear apertures has been used to

formulate expressions for the phase error caused by displacing the feed and to illustrate

the pattern effects of the error. [37, 19] For an axisymmetric parabolic reflector with a

feed at the focus, the far-field is (see Fig. 3.1-1) [37]

2_ a

E(O, ¢p) -- ; _ fCr, q_' )eJkCp- (3.1-1)

0 0

where a is the radius of the aperture, f(r, _') is the effective aperture distribution, p

defines the parabolic surface as p = 237(1 -cos 0') , /_ is a unit vector in the direction

of/_, • is the perpendicular distance from the z-axis to the point of integration on the

reflector surface, and the constants have been suppressed. Assuming that for small feed

displacement thef(r, _b' ) is nearly unchanged, the far-field expression becomes [37]

2_ a

0 0

f (r, dp' )e Jk( ?'- _'" R)rdrdc_' (3.1-2)

where

= + + (3.1-3)

2 e x _b' 0' 2 e z

1 + ----7 cos sin + --7- cos 0' + 2

ex + _z

p2

(3.1-4)

3.0 REFLECTOR SCANNING 83
X

P

m

e' R

Figure 3.1-I. Geometry of the axisymmetric reflector used to analyze the errors

associated with the displaced feed. [37]

3.0 REFLECTOR SCANNING 84
3.1.1 PHASE ERROR TYPES

For small feed displacements [37]

_JC

-'7ex <"7 < < 1 (3.1-5)

the phase term in (3.1-2) can be approximated by (neglecting terms higher than O(e_))

[37]

p' - _' • R = 2f- e x cos _b sin 0 - ez cos 0

- p sin 0' sin 0 cos(_' - 4,) + ex cos _b' sin 0'

2

_x

+ ezcos0' +.-_-p +pcos0'(1 --cos0)

2

- _ c°s2_b' sin20 '

(3.1-6)

where the In'st three terms are constants with respect to the integral of (3.1-2) and can

be taken out of the integral. The next term is the normal phase term of a smooth

parabolic reflector with the feed at the focus (see (3.1-1)). The remaining terms represent

the phase errors introduced by displacing the feed from the axis.

The fifth term in (3.1-6) represents phase errors that are linear and cubic as a

function of x. This is now demonstrated. Since [371

sin/_'= r[f rIl_( r )2 r )a 1

1 + (r12f) 2 = f "_ + (_ .... (3.1-7)

the phase error associated with the fifth term may be written as [37]

3.0 REFLECTOR SCANNING 85
[ ,, +(V,),.... ] (3.1-8)

where [37]

e X

us=--f-ffi tan O, (3.1-9)

is a measure of the feed squint. The first term in (3.1-8) is linear as a function of x,

x = • cos _'. It is represented graphically in Fig. 3. l-2(a). This will cause a pattern shift

equal to the feed squint away from boresight. [37] The second term in (3.1-8) is cubic

term known as the coma aberration. It is shown in Fig. 3.1-2(b). It has perhaps the

most severe impact on the radiation pattern of all the phase error terms. It causes a

beam shift in the opposite direction of the linear term and degrades the pattern. [37] The

pattern effects of just the cubic error term are shown in Fig. 3.1-3. The patterns in Fig.

3.1-3 are for a one-dimensional aperture. There is an additional exponential term in the

far field pattern expression that represents the cubic error. The far field pattern for the

cubic error is (following [19])

1

=2 f (3.1-10)

-1

where d is the diameter, f(x) is the aperture distribution, u = (rta/2) sin 0, and, for the

cubic error, AL = Jx _. The remaining terms in (3.1-8) are higher order coma terms. For

high F/D-systems, they cause negligible contribution to the distortions. [37]

The sixth through eighth terms in (3.1-6) are field curvature terms (functions oft 2)

[37] which are represented graphically in Fig. 3.1-2(c). The effects of this quadratic

phase error are lower gain, higher side lobes, and f'dling of the nulls. [19] Figure 3.1-4

3.0 REFLECTOR SCANNING 86
//I

M.

(a) (b)

1

\ j

Y

ad.

/

I j

(c) (d)

Figure 3.1-2. Phase errors associated with a small displacement of the feed from the

Focus. The phase errors, AL, are shown for the cases (a) linear, (b) coma,

(c) curvature of field, and (d) astigmatism. (From [19])

3.0 REFLECTOR SCANNING 87
3T

Figure 3.1-3. Pattern degradation

one-dimensional aperture.

due to the coma phase error only for

For this figure, the error is AL = <_x3 . [19]

3.0 REFLECTOR SCANNING 88
shows the effects of just the field curvature terms. Equation (3.1-10) was used to

compute the far field patterns of Fig. 3.1-4 with AL =/_x 2. [19] The error is symmetric

with respect to the reflector axis so there is no scanning due to the curvature of field.

The field curvature may be eliminated by axially displacing the feed parallel to the z-axis

to a location corresponding to the Petzval surface from optics. [37] It is found by

setting the field curvature terms to zero and solving for the axial displacement [37]

2

Ex

2: (3.1-1l)

Past studies indicate that the optimum z-displacement for maximum gain actually lies

between the focal plane and the Petzval surface. [32, 12] This could be due to the loss

from the field curvature phase error terms being overcome by the decrease in spillover.

[321

The last term in (3.1-6) is the astigmatism. It causes an asymmetric distortion in

the radiation pattern. The astigmatism is shown in Fig. 3.1-2(d). The effects of the

astigmatism are negligible for most parabolic antennas. Following [37], the ratio of the

astigmatism to the total coma at the edge is [37]

Astigmatism 2us( f]2a)

Total coma [ 1+ (a/2f)2]2

(3.1-12)

which, for small displacement e,, is a small quantity and the coma aberration effects

would dominate.

3.0 REFLECTOR SCANNING 89
NO PHASE ERROR

Figure 3.1-4. Pattern degradation due to the field curvature phase error only for a

one-dimensional aperture. For this figure, AL =/_x 2 . [191

3.0 REFLECTOR SCANNING 9O
3.1.2 BEAM DEVIATION FACTOR FOR AXISYMMETRIC

REFLECTORS

The scanning of a reflector is often characterized by the beam deviation factor

(BDF). The BDF is essentially a measure of the equivalent curvature of the reflector

surface. [54] The BDF is defined as [19]

BDF= O-''2-B (3.1-13)

0t=

where Oa and 0r are the beam scan angle and the feed tilt angle as defined by Fig. 3.1-5.

For small feed displacements, the position of the beam maximum for the far field pattern

is the value of u = sin 0 that minimizes the illumination-weighted squared phase error.

[37] The minimization of the weighted error of a circularly symmetric aperture

distribution leads to the classic result (using the geometry in Fig. 3.1-1) [37]

f(r)r3 dr

1+ (r/2f) 2

BDF = o

a

f f(r) r_dr

o

where f(r) is the aperture distribution.

with the feed on the Petzval surface.

small feed displacement is given by [20]

BDF =

I + k(D/4f)2

l + (d/4f)2

(3.1-14)

Equation (3.1-14) is valid for small scan angles

An approximate formulation for the BDF for

(3.1-15)

3.0 REFLECTOR SCANNING 91
D F

FEED

Figure 3.1-5. Angle definitions for the BDF of a symmetric paraboloid. The angle

Op is the angle measured from the axis passing through the beam directed

in the On-direction. [19]

3.0 REFLECTOR SCANNING

92
where the diameter of the aperture D = 2a, and k is an empirically determined constant

such that 0 < k < 1. The values for k are generally in the range 0.3 < k < 0.7 and k

becomes larger with higher aperture tapering. [20]

The scan limit of a reflector is often chosen to be the angle where the gain drops

by 1 dB. Ruze's approximate expression for the scan limit using the 1 dB loss criteria

is [371

N s = 0.44 + 22(F/D) 2 (3.1-16)

where Na is the scan limit in beamwidths. This expression is limited to small angle scans.

Mrstik [24] developed a numerical approach using perturbation analysis in which

the gain loss of the scanned beam was related to the rms phase errors introduced by

displacing a single feed. [2al The numerical method is valid for any scan angle. Figure

3.1-6 shows plots of the numerical approach compared to (3.1-16). The curves for

(3.1-16) give somewhat optimistic values for the scan limit in wide angle scan regions

where the approximation used to derive (3.1-16) fails.

Examples of scanned beams are shown in Fig. 3.1-7 for a reflector with F/D = 0.4.

Note that as the pattern is scanned farther from boresight, the distortion in the pattern

increases. The first large side lobe on the axis side of the main beam is known as the

coma lobe. The main lobe broadens with increasing scan and the first side lobe on the

other side decreases, changes sign, and merges with the main lobe and second side lobe.

[37]

The effects of the phase errors can be decreased somewhat by using a higher F/D

reflector. This allows for scanning at wider angles. Figure 3.1-8 shows scanned patterns

for a reflector with F/D ffi 1.0 . Note the improved performance for the higher F/D

reflector as compared to the patterns in Fig. 3.1-7. The drawback to using a higher

3.0 REFLECTOR SCANNING 93
10

58

,N

O

6

m

O 4

i

0

lo2

\_ _\"_ _ _' _ _ ------ NEW RESULTS

\\ \\ \\

\\- \ "\\ \_ \1o

N _ 4

2 _'

Io3 io4 1o5 1oe

NORMALIZED APERTURE DIAMETER (D/X)

Figure 3.1-6. Scan limit comparing Ruze's approximation from (3.1-16) [37], the

solid curves, and Mrstik's numerical approach [24], dashed curves, for a

gain loss criteria of 1 dB. Ruze's approximation fails for wide angle scans

where the approximations used to derive (3. I-16) fail. (Note: the "reflector

F-number" is the F[D ratio) [24]

3.0 REFLECTOR SCANNING 94
m -IC

U

u.I

 -2o

- 16 - 12 - 8 - 4 0 4 8 12

ka ll_ e

-6O

- 20 16 20

(a)

8 12 16 20

(b)

Figure 3.1-7. Radiation patterns for an axisymmetric (F/D = 0.4) reflector with a

laterally displaced feed. The patterns are for scans of (a) 0 beamwidths,

(b) 2 beamwidths, and (c) 6 beamwidths. The angle 9p is measured from

0n (see Fig. 3.1-5). [19]

3.0 REFLECTOR SCANNING 9S
- I0i

-6O

-20 -16 -12 -8 -4 0 4

kasln 0

P

(c)

\

V 12 16 20

Figure 3.1-7. (cont.)

3.0 REFLECTOR SCANNING 96
! -1o

I -20 |

ua

-6O

-20 - 16 - 12 -8 -4 0 4

kaMn

(a)

8 12 16 20

141

el

I

|

0

-I0

-20

" r" _-:_ _ _oaf 

-50

-60

-20 - 16 - 12 -8 -4 0 4 8 12 16 20

Im .a-, _e

(b)

Figure 3.1-8. Radiation patterns for an axisymmetric (F/D = 1.0) reflector with a

laterally displaced feed. The patterns are for scans of(a) 8 beamwidth and

(b) 16 beamwidths. Note the improvement over Fig. 3.1-7. The angle

0p is measured from 0_, (see Fig. 3.1-5). [19]

3.0 REFLECTOR SCANNING 9'7
F/D reflector is that it requires a larger support structure for the feed. In addition, it

requires a larger feed displacement to scan a given angle than lower F/D reflectors.

3.1.3 BEAM DEVIATION FACTOR FOR OFFSET REFLECTORS

The above discussion on displaced feed scanning pertained to axisymmetric

reflectors. Offset configurations are often used to overcome blockage problems. The

effects of the phase errors on the scanned patterns of offset reflectors are better

characterized by the F/Dp (focal length to diameter of the parent paraboloid) ratio,

rather than the actual F/D ratio. [19] The BDF of an offset reflector is computed using

the angles defined in Fig. 3.1-9. Equation (3.1-13) is still valid for the angle 0r measured

relative to the ray locating the center of the reflector (the ray locating the center of the

dish passes through the focal point along the angle _kc) and Oa located relative to

boresight.

3.2 EL EC TR ONIC SCANNING

Phase errors are the primary cause of distortion in the scanned radiation pattern

of a reflector with a single displaced feed. The effects of the phase errors can be

somewhat overcome by using an array of feeds. In a receive mode, the array elements

are used to capture energy spread over the focal region and coherently combine it.

The studies using array feeds are divided into two main topics. Fixed array feeds

are used for small angle scanning of the reflector without mechanical movement.

3.0 REFLECTOR SCANNING 98
D

L

Figure 3.1-9. Angle definitions for the BDF of an offset paraboloid. [19]

3.0 REFLECTOR SCANNING 99
Movable array feeds are used for wide angle scanning and are located at some optimum

point for a given scan angle.

3.2.1 FIXED ARRAY FEED SCANNING

Of all the configurations, scanning with a fixed array feed is probably the most

desirable from a system standpoint since there are no moving parts. A reflector with a

fixed moderate size array feed is capable of scanning over a small number of

beamwidths. To increase the scanning range significantly would require a very large feed

array.

The majority of fixed array feed reflector scanning is accomplished using the

method of conjugate field match (CFM) to compute the excitations for the feed elements

(see Sec. 2.3.2). The CFM excitations are the complex conjugates of the field received

by the ith array element as a result of a plane wave incident on the reflector aperture

from the desired direction of scan. [7] The CFM excitations coherently combine the

energy and increase the gain. [7] The program SCANI_,P (see Sec. 5.2) uses CFM to

compute the excitation coefficients for a scanned reflector with an array feed.

Studies were performed by Assaly and P-,icardi [3] and Mrstik and Smith [23] on

infinite cylindrical reflectors to examine the feasibility of scanning with fixed array feeds.

The simpler math of the cylindrical reflectors provides better physical insight into the

problem.

A moderate size cylindrical reflector with a fixed array feed was analyzed to

examine the parameters associated with the scanning characteristics. [3] The

axisymmetric reflector (F/D _. 0.5, D -- 60_l) had an array feed with various numbers of

elements located a distance R from the parabola vertex and in a plane parallel to the

3.0 REFLECTOR SCANNING 100
focal plane (see Fig 3.2-I). The feed array elements were symmetrically located at a

distance o from the focus-vertex plane. The o ranged from -6). to +6).. DCFM (See.

2.3.2) was used to compute the feed excitations. The reflector is assumed to be

illuminated by a plane wave incident from the desired direction of scan. The complex

conjugates of the received signal are used as the DCFM element excitations.

The reflector was scanned from boresight out to 00 = 8 ° • Three cases were

evaluated for 21, 16, and 13 elements spaced 0.6)., 0.8)., and 1.0,_, respectively, with

R = 242. Figure 3.2-2 shows examples of scanned patterns. Note that for Figs. 3.2-2(b)

and 3.2-2(c) there are large side lobes. These are due to the larger spacings between the

elements. The large element spacing does not allow the array to properly match the

focal plane fields of the reflector. The minimum spacing providing satisfactory

performance was found to be about 0.7). . [3]

If the analysis were extended to a three-dimensional problem, the array would have

to be increased considerably; the number of elements is n 2 instead of n elements as in

the linear array feed for a cylinder. [3] For example, instead of 21 elements as in Fig.

3.2-2(a), about 441 elements would be required.

Mrstik and Smith [23] analyzed very large infinite cylindrical reflectors to determine

the feasibility of scanning many beamwidths off boresight with fixed feed arrays.

Electronic scanning analysis of these very large reflectors shows that wide angle scanning

is not feasible with fixed feeds and further illustrates the problem of requiring unrealistic

numbers of feed elements. The pattern results are discussed in Sec. 3.2.2.

One of the reflectors considered in [23] was axisymmetric with

F/D= 1.0, D= 1000). To continuously scan the cylindrical reflector +5 ° (:t:87

beamwidths) would require a feed array approximately 229). in diameter with 0.5).

spacing (458 elements). To extend this to a three-dimensional problem would require

on the order of 200,000 elements.

3.0 REFLECTOR SCANNING 101
_ _ "_ w _

/ \

I

(o,o ..... I

I

/

\

\ /

\ /

\ /

f jt_ __

Figure 3.2-1. Geometry of the reflector used by Assaly and Ricardi to analyze the

rLxed array reflector scanning. [3]

3.0 REFLECTOR SCANNING 102
(a)

g

*Z -8 -a _ 4 8 '2 -8 -4

:.! I! r".-

_, ANGLE OFF BORESIGHT (deq}

(b)

(c)

Figure 3.2-2. Far-field patterns for a cylindrical reflector (F/D = 0.5, F = 30).) with

a linear array feed located R = 24). from the parabola vertex with array

feed diameter 122 and (a) 21 elements with spacing of 0.62, (b) 16

elements with spacing of 0.8)., and (c) 13 elements with spacing 1.0).. [3]

3.0 REFLECTOR SCANNING ORIGINAL PAGE IS

OF POOR QUALITY

103
Electronic scanning with moderate size fixed array feeds was studied by Blank and

Imbriale [4, 7] for paraboloidal reflectors. ICFM (see Sec. 2.3.2) is used to determine the

element excitations. [7] The reflectors (F/D--1.0) were assumed to be smooth or

distorted with [7]

_. _ e p"cos(L¢ ) (3.2-1)

where Az is the deviation in the z-direction of the distorted reflector from a smooth

reflector, a is the maximum deviation (,_), s is a real number, p is the normalized distance

from the z-axis to a point on the reflector, L is the integer number of periodic scallops

on the reflector, and _ is as defined in Fig. 3.2-3(a). The feed array was chosen to be a

triangular grid with N-7, 19, 37 elements with optimal element diameters designed

to maximize the gain on axis. [7] The optimal element diameters were determined

empirically by evaluating the data from a parametric study. [7] Figure 3.2-3(b) shows

the array configurations.

The curves in Fig. 3.2-3(c) show the gain loss as a function of scan angle for various

numbers of feed elements and different distortion cases. The curves show that for the

optimum gain 7-element case, the spacing of the feeds is too large to allow continuous

scanning of a beamwidth. The 19-element case provides continuous scanning over +1

beamwidths while the 37-element case extends the continuous scanning to :]:1.8

beamwidths. [7]

3.2.2 MOVABLE FEED ARRAY SCANNING

Scanning a reflector with a movable array feed is not as desirable as the luted array

feed since it requires a moving feed assembly. It does, however, use a much smaller

3.0 REFLECTOR SCANNING 104
/

¥

/ /

I

/ /

(a)

/

\

:/

:/

/

(b)

Figure 3.2-3. (a) Reflector geometry for scanning a paraboloida! reflector with a

fixed array feed. (b) Optimum gain array reed configurations. (c) Gain

loss as a function orscan angle (F/D= 1, s= 2, e --0, 0.12, 0.2). [7]

3.0 REFLECTOR SCANNING ORI_L_,_AL PAGE IS

OF POOR QUALITY

105
/\

0.0

SCAN ANGLE. BW

(c)

25

Figure 3.2-3. (cont.)

3.0 REFLECTOR SCANNING

I06
array when wide angle scanning is required. The pattern of a reflector using a movable

array feed is, of course, much improved over that of a single displaced feed.

Rudge and Withers [38] developed a double Fourier transform technique for

scanning a finite cylindrical reflector with a movable array feed. The technique exploits

the approximate Fourier transform relationship between the received aperture plane

distribution and focal plane fields. For a plane wave received at boresight, the received

aperture plane distribution is a constant amplitude, constant phase function and the

focal plane fields are of the "sin _x/_.x" form. If the focal plane field is acted upon by

a device that implements another Fourier transform, the output of the device is again a

constant amplitude, constant phase function. In practice, the second Fourier transform

has been approximately implemented with a Butler matrix. [38]

$

The reflector subtends an angle 20 (see Fig. 3.2-4(a)). Rudge and Withers [38]

found that as a single feed is moved if it maintains the subtended angle constant 20

(defined in Fig. 3.2-4(b)), that the relationship between the aperture plane fields and the

focal plane fields as defined in the new transform plane are related by a Fourier

transform as in the unscanned case. For a plane wave received in a direction other than

boresight, the plane wave is acted upon by a second Fourier transform in the new

transform plane. The output is a constant amplitude function with phase variation that

is the complex conjugate of the aperture plane field. [39] Phase shifters can then be used

to coherently combine the received signal and feed element amplitude control is not

necessary. [38] This method was experimentally implemented with a moderate size

cylindrical reflector and scanning of +15 beamwidths was achieved. [38]

Hung and Mittra [12] used the CFM method to improve the scanning

characteristics of a reflector with a movable array feed. The analysis involved designing

the geometry of a moderate size feed array based on the focal region field distribution.

3.0 REFLECTOR SCANNING 107
APERTURE /

PLANE F/

YyTRANSFORM PLANE E

INCIDENT WAVEFRON T

(a)

APERTURE /""l

PLANE F

INCIDENT

WAVEFORM

(b)

Figure 3.2-4. Location of the transform planes For a reflector that is (a) unscanned

(focal plane) or (b) scanned off boresight. [39l

3.0 REFLECTOR SCA.NNING 108
A large axisymmetric paraboloidal reflector ( F/D -- 0.48, D = 5502 ) was evaluated

for wide angle scanning. The focal region field contours are shown in Fig. 2.2-9 for the

cases of a received plane wave incident from 2.5 ° and 5 °. From evaluation of the focal

plane fields, a 27-element RN _ triangular array of Fig. 3.2-5(a) was chosen to feed the

reflector. [12]

The array center was located in a position that maximized the gain for a given scan

location. [12] This optimum location corresponded to the peak of the focal region field

distributions from Fig. 2.2-9. The secondary pattern for the cases of no scan, 2.5 ° scan

(24 beamwidths), and 5.2 ° scan (48 beamwidths) are shown in Fig.'s 3.2-5(b), (c), and

(d), respectively. The scanned pattern produced by the array is compared to that of a

scanned pattern produced by a single feed.

Directivity optimization has also been used to improve the scanning characteristics

of an offset reflector with a movable array feed. [17] Directivity optimization is a

method for computing the feed array element excitations that maximize the directivity

of the reflector for a given angle of scan. The formula for directivity of a reflector with

an array feed was differentiated with respect to the unknown excitations of the array

elements. The resulting equation was then solved for the excitations that give optimum

directivity.

The reflector that was analyzed with directivity optimization was an offset reflector

(F= 94.87A, D = 108.15,_, F/Dp -- 0.38, Fig. 3.2-6(a)). [17] For a 10 beamwidth scan, the

feed was displaced by 10.653. in the y-direction. Three patterns are plotted in Fig.

3.2-6(b). The results are shown for a single displaced feed and for 19-element triangular

grid arrays with either CFM or directivity optimization excitations. As in most of the

examples in the paper, the directivity optimization produced patterns with the highest

directive gain. [17]

3.0 REFLECTOR SCANNING 109
/

o

!

|

|

|

|

llO

3.0 R_,FL_crrOR SC_X_I_G
(SP) ,_lJAllD3810 70_-3DN3_3338

bJ

rr

W

O

c_O

v

°_

L

0 ,_

•_, ._

E

"J..j _

_'_

.., _'_

E "_=

L--

o "_.E-

_o, __

oh.

_..--

i

e_

_4

L

._

3.0 REFLECTOR SCANNING 111
3.3 DUAL REFLECTOR SCANNING

In Sec. 3.1, it was pointed out that the effects of the phase errors associated with

scanning a reflector can be decreased somewhat by using a higher F]D reflector. To

increase the F]D of a single reflector would require a larger feed structure. A possible

alternative is to use a dual reflector with a higher virtual F]D.

The scanned dual reflectors considered for this study are of the classic Cassegrain

or Gregorian types, The subreflector is either a hyperboloidal or an elliptical function

and the main reflector is paraboloidal. Shaped reflectors were not considered.

Early Cassegrain reflectors were analyzed by the concept of an equivalent parabola.

[13] This technique is advantageous as it saves much computational time since only one

reflector needs to be analyzed. The main reflector and subreflector are replaced by an

equivalent parabolic reflector with focal length F, (see Fig. 3.3-1). Some important

parameter relationships between the dual reflector and the equivalent reflector are [I 3]

1 D,_ 1

fi _ =tan-_-,,

(3.3-I)

x, = -_- (3.3-2)

Fe tan + *v Lr e + 1

:I:"b'_"m= tan+,, :"_"= e-I (3.3-3)

where in (3.3-3) the positive sign is for the Cassegrain and the negative sign is for the

Gregorian.

3.0 REFLECTOR SCANNING 112
E

Q

/

/

PRINCIPAL SURFACE AND EQUIVALENT --\/PAIABOLA --/

.../_ ,=

/

Fe

Figure 3.3-1. Geometry of a Cassegrain reflector and its equivalent parabolic

reflector. [13]

3.0 REFLECTOR SCANNING

113
The pattern of a Cassegrain can be evaluated using the equivalent reflector

technique for small angle scans. The coma aberration of the Cassegrain is nearly the

same as for the equivalent reflector when the feed is displaced near the focus. [13] For

larger scans, however, the equivalent parabola concept fails to provide accurate results

[40]; this is discussed further now.

Wong [59] compared the equivalent reflector technique to a PO/PO analysis of a

Cassegrain reflector for small scans. Figure 3.3-2(a) shows the geometry of the PO/PO

analysis. The feeds were assumed to have a pattern approximated by a cos"0-type power

pattern. [59] The results of the study for various values of n and ratios of the

subreflector diameter and main reflector diameter showed that the aperture efnciencies

of the PO/PO analysis and the equivalent reflector analysis differed by less than 10%.

[59] Examples of scanned radiation patterns comparing the two techniques are shown in

Fig. 3.3-2(b). The study concluded that the equivalent parabola technique was valid for

up to about 4 beamwidth scans. [59]

Rahmat-Samii and Galindo-lsrael [40] evaluated the equivalent paraboloid

technique for offset Cassegrain reflectors using GTD/PO analysis of the dual reflector.

[40] The gain loss as a function of the number of beamwidths scanned is shown in Fig.

3.3-3. The plots in Fig. 3.3-3 compare the characteristics of the Cassegrain, the

equivalent paraboloid, and the main reflector by itself. The plots show that the scan

performance of the actual dual reflector is somewhere in between that of the equivalent

paraboloid and the single main reflector. [40]

Optimum feed locations for prime focus reflectors were related to the Petzval

surface in Sec. 3.1. Krichevsky and DiFonzo [15, 16] studied optimum feed locations for

Cassegrain and Gregorian antennas. The method for finding the optimum feed locations

has two steps. The locus of feed positions that produce a scanned beam in the desired

direction is determined with an approximate ray tracing technique. [15] The fLrSt order

3.0 REFLECTOR SCANNING 114
lip NI M YO

f

i; °-*

\\

o

o

_,w,.

v

3.0 REFLECTOR SCANNING

ORIGINAL pAGE IS

OF poOR QUAL|T_

¢1

,J=

I-=

O

¢)

t.=

_jw

_._o_

e.j

_.>

v

e.j

O.,1,

¢_,,= M

e-.

_...=

_..-j

Pql_

¢et

==

115
o, o**.,o,.

-02 '"'-

-0.<

-O.i

-0.I

$ -I0

-1.2

-1.4

-1.6

0

EQUIVALENT

F,/D ,-0 6

%..

\ \

\ i

\

\

PARABOLOID k

F/Dp - 0.4 \

\

PARABOLOID

.

%°°°°°

%.

%%

%%

CASSEGRAIN

"115

'/\

\

2 3 4, 5 6

BEAMI_'Ir"r_ SCANNED |BlWlWS)

Figure 3.3-3. Gain loss performance for an offset Cassegrain reflector. The plots

show the loss characteristics for the Cassegrain reflector using GTD/PO

analysis (solid curve), equivalent paraboloid (dotted), and prime focus

system using the main reflector (dashed). [19]

3.0 REFLECTOR SCANNING i!6
approximation of the locus of points for a constant beam scan angle is a straight line

(valid for small angle scans). [15] The optimum location for the feed is at the point along

the locus that minimizes the phase error for the given direction of scan. [15] Computer

analysis of several antenna configurations showed that this method provides the best

location for the feed in terms of gain, pattern symmetry, and null depth. [16]

The geometry of an offset Cassegrain reflector is shown in Fig. 3.3-4(a). To a first

order approximation, the optimum feed loci of a Cassegrain reflector for given scan

angles are located along a line. [15] In Fig. 3.3-4(b), the dashed line represents the first

order approximation to the loci of optimum feed locations. To a second order

approximation, the optimum feed loci of a Cassegrain reflector are located along a

hyperbola. [15] In Fig. 3.3-4(b), the solid hyperbolic curve represents the second order

approximation to the loci of optimum feed locations. For a Gregorian reflector, the

second order approximation places the optimum feed loci along an ellipse. [16]

Examples of scanned beams as a function of several feed locations for the offset

reflector are shown in Fig. 3.3-5. The radiation patterns of the reflectors with their feeds

located along the optimum second order curve have the best characteristics.

3.0 REFLECTOR SCANNING 117
o

0

r_

m

_o

i,,-

/

/'

S I

tY

G.

i

s

s

3.0 REFLECTOR SCANNING

,.m

v

@

O

P.

r-.

• _ r..e

=8

C _

L--

llS
"D

o I

5 -

10

15

20

25

30

35

40

45

5O

/

j:"i

/

IO\

,%

4

;\

,I_[DEG)

(a)

rn

°I,, \t; '

5 4 "/

15k /

/ /

I- ' ?

L,li /.

i'k,'li'77'

.olTv 45

-50 .... .i_ __ i____

I-,--4

,'! 7' _ ",,"-_ I

'S \'-°' ' 1

7 t\ i .4 i

• (D{Gi

(b)

Figure 3.3-5. Patterns for the offset Cassegrain reflector as l'unctions oF feed

location, The leeds arc located in two arbitrary locations and at the

optimum locations for the first order and second order approximations.

The patterns arc for beams scanned to (a) -4 ° (first order - point 3; second

order, point 2) and (b) + 4 ° (fLrst order - point 2; second order - point

3). [15l

3.0 REFLECTOR SCANNING

oRIGINAL PAGE IS

OF POOR QUALITY

119
4.0 PATTERN SYNTHESIS COMPENSATION

TECHNIQUE

w

The pattern synthesis compensation technique is based on the iterative sampling

method (ISM). The ISM was originally formulated to synthesize shaped radiation

patterns for line sources, linear arrays, rectangular apertures, and rectangular arrays. [48,

49, 50, 51] The pattern synthesis compensation technique corrects pattern irregularities

caused by surface errors by superposing correction beams with the original distorted

pattern. The corrections can be applied in a series of iterations.

In this chapter, the original formulation for ISM is outlined. The ISM is extended

for application to reflector problems. This extended ISM is then applied to formulate

the pattern synthesis compensation technique.

4.0 PATTERN SYNTHF.SIS COMPENSATION TECHNIQUE 120
4.1 ITERA TIVE SAMPLING METHOD

4.1.1 ITERATIVE SAMPLING METHOD ALGORITHM

The iterative sampling method for real-valued beam synthesis begins with some

original pattern that is an approximation to the desired radiation pattern. The first

iteration is to add a series of correction patterns to the original approximate pattern.

If the resulting pattern is not satisfactory, more iterations of correction patterns are

applied. The total pattern correction is the sum of the weighted correction patterns.

For the ith iteration, the total correction for a linear source is [48]

,_F<o(_,) =Z_ (°aCu- _,(o) (4.1-I)

n

where , for a z-directed source, u = cos 0, _o is the correction pattern weight, and

G(u-_o) is the normalized pattern with a main beam centered at u_0 and maximum

amplitude of unity. After K iterations, the resultant improved approximation to the

desired pattern is [481

g

FifO(u) -- F (°)(u) + _"_AF (0(u) (4.1-2)

l-I

where F_°>(u) is the original approximate pattern.

The correction pattern for a line source has a correction current _'_(s) related to the

pattern by [481

4.0 PATTERN SYNTHESIS COMPENSATION TECHNIQUE 121
L_

(4.1-3)

where s = zlJl and L_, is the length of the source.

ith iteration for the line source is [48]

The total correction current for the

(4,1-4)

and the total current after K iterations is [48]

f_nO(s) -- f(_°)(s) 4- Z_f_")(s) (4.1-5)

I

wheref_,_(s) is the current distribution of the original approximate pattern.

For a linear array, the corresponding correction current terms g_ are related to the

correction pattern by [48]

m

(4.1-6)

where s, are the normalized element positions. The total correction for the ith iteration

current of the ruth element of the linear array is [48]

A/(m0 k" (0 (0 =/_.,ang_n (4.1-7)

and the total current of the ruth element aRer K iterations is [48]

4.0 PATTERN SYNTHESIS COMPENSATION TECHNIQUE 122
-- + (4.1-8)

where I_ is the current excitation for the original approximate pattern.

A good choice for the weights is [481

..+= rd(._o) - F<_-,>(_o ) (4.1-9)

where F2u) is the desired pattern. A good choice for the iterative sample points is [48]

• (t-l) , q-l) ..<o "_ +"_-, (4.1-I0)

un 2

As the number of iterations increases, the pattern approaches the desired pattern to

within some acceptable limits and the number of iterative sample points decreases. [48]

The two-dimensional aperture or array problem has a similar formulation except

that the single integrals and summations now become double integrals and summations.

The patterns and distributions are now bivariate. [501

4.1.2 APPLICATION OF ISM TO REFLECTORS

In this section, a formulation for extending the ISM to reflectors is presented.

Implementing the iterative sampling method with reflector antennas is a difficult

problem. With a planar phased array, there is much more control over the amplitude

and phase of the radiating currents. With a reflector, the radiating current is an induced

current on the surface of the reflector. Scanning of correction patterns is accomplished

with feeds displaced from the focal point or by an array of feeds weighted to

4.0 PA'FrERN SYNTHESIS COMPENSATION TECHNIQUE 123
electronically scan beams. The amplitude and phase of the scanned pattern currents

have inherent amplitude and phase errors that cause the side lobe level and beamwidth

to increase with increasing scan. The gain of the scanned correction patterns decreases

with increasing scan (see Chapter 3). Furthermore, the radiation pattern is in general a

complex valued vector quantity so the phase of the pattern at the iterative sample points

is important.

Figure 4.1-1

y-polarized feed.

cylindrical reflector is

shows the geometry for

For the ith iteration, the

an infinite cylindrical reflector with a

total correction pattern of an infinite

AF<0Cu) = Ea(n 0 G,el(U, u(n0 ) (4.1-I I)

tl

where for this geometry u= sin 0 (not cos 0 as for the z-directed sources) and the

u- 4 o from (4.l-l) has been replaced to emphasize that for the reflector, the radiation

pattern is no longer a shifted universal pattern, as is often the case for linear arrays or

line sources. The u_o is where the main beam of the nth correction pattern for the

reflector is centered. The reflector is assumed to be y-polarized. After K iterations the

resultant improved pattern is

K

F (_(u) = F (0) + EA F (O(u) (4.1-12)

1==1

The correction pattern for the cylindrical reflector is related to the current g_2, by (see

Ch. 5 for more details)

so,

r?l

s,

4.0 PATTERN SYNTHESIS COMPENSATION TECHNIQUE

(4.1-13)

124
So

g(O

mn

m

Hinc

ruth

J

I

I

I

J

feed

_4

R 1

v

Z

Figure 4.1-1. I SM cylindrical reflector geometry.

4.0 PA'FFERN SYNTHESIS COMPENSATION TECHNIQUE

125
where _¢ is a constant and/t,_ is the H-field incident upon the reflector from the ruth

feed, _ locates integration points on the reflector surface, and R, is a unit vector in the

direction of the observation point. For displaced feed scanning (Sec. 3.1), the g<_ in

(4.1-13) are all zero except the one that produces the nth scanned beam. For electronic

scanning (Sec. 3.2), g<_ is the induced current of the ruth feed that has been amplitude

and phase weighted (for example, using CFM, Sec. 3.2) to compositely produce the nth

scanned beam when superposed with the induced currents from the other feeds. The

total correction current contribution for the ith iteration of the ruth feed is

(0 v, (0 (0

Afr,f_ = /_.,ah gf, n (a.l-14)

and the total current on the reflector surface for the ruth feed after K iterations is

The expression for a_O is similar to (4.1-9). Figure 4.1-2 shows the elements of the

nth correction beam weighting scheme for the ith iteration at up . To allow for an

unnormalized correction pattern, the expression from (4. l-9) needs to be divided by the

magnitude of G,,/(u, up ). Assuming F_(up) to be real and positive, the real-valued

amplitude of a_o is

The a_ can be positive or negative (the phase ofa_,o is 0* or 180"). The correction beam

must satisfy an in-phase or anti-phase condition with F_-Z_(u_0 ) for correction at u_0. The

4.0 PA'IWERN SYNTHESIS COMPENSATION TECHNIQUE 126
I \

F(;-')Cu_))

f \

\/ N/ / L,I

Figure 4.1-2. Representation of elements used in the ISM weighting scheme for the

ith iteration at _'_. The F<'-*_(u<_ ) ) is the corrected pattern at the i - 1 step.

G,,l(u, _'_) is the correction pattern with the peak at u_'_.F_(u<;)) is the

desired corrected pattern level at u_'_.

4.0 PATTERN SYNTliESIS COMPENSATION TECHNIQUE 127
phase of F_'-_)(up) must then be equal to or opposite the phase of the weighted

correction pattern a_G,,i(u, u_). Therefore,

[0.] _G+ _'.+ ='_p (4.1-17)

:1: 180"

and, solving for the weight phase,

¢_"='¢_F-¢_°-[ j+180" O°

(4.1-18)

where $o, St, and $, are the phases of G,,/(u, up ), F_,-,,(M0) , and the complex part of

a_o, respectively. From (4.1-16) and (4.1-18),

a(nO = =_ e J_= (4.1-19)

The three-dimensional reflector problem has a similar formulation except that the

single integrals become double integrals and the patterns and current distributions are

now bivariate.

The increasing side lobe level and beam broadening are the fundamental limitations

to implementing the ISM with reflectors. If several correction patterns are applied in a

given iteration, the resultant pattern will equal the desired pattern at the sample points

only if the correction patterns are uncorrelated at the sample points. [50] Good results

can be obtained only if the correlation between the correction beams is weak at the

sample points. The ISM for reflectors is therefore limited to an angular extent over

which the side lobe level increase and beam broadening do not cause significant

interference between pattern corrections.

4.0 PA'VrERN SYNTHESIS COMPENSATION TECHNIQUE 128
4.2 PA TTERN SYNTHESIS COMPENSATION

TECHNIQUE DESCRIPTION

The pattern synthesis compensation technique is essentially the ISM applied to

moderately distorted reflectors. Surface errors cause increased side lobe levels (see Sec.

2.1). The ISM can be used to reduce the side lobe levels.

The pattern synthesis compensation technique differs from most compensation

methods in two ways (see Sec. 2.3). It is a localized algorithm in which only the parts

of the radiation pattern that do not meet desired specifications are compensated. Most

of the other techniques try to use array feeds to correct the focal plane fields near the

focal point and, hence, compensate for the distortions of the overall radiation pattern.

The other difference is that the pattern synthesis technique does not require reflector

surface data. Radiation pattern data are required. If surface data are available,

however, the pattern data that are required can be computed.

The correction beam scanning is implemented in two ways. Displaced feed

scanning is accomplished by moving auxiliary feeds away from the focal point. For

displaced feed scanning, the g¢_ of (4.1-13) would all be zero except for the feed that

corresponds to the nth correction beam. Superposition is performed with the secondary

radiation patterns of the individual feeds. This is the easiest analytical implementation

of the pattern synthesis compensation technique. Electronic beam scanning with a fixed

array feed is a more desirable implementation and probably more practical from a

mechanical standpoint. The g2, of (4.1-13) correspond to the induced current of the

ruth feed that has been amplitude and phase weighted (for example, using CFM, Sec.

3.2) to compositely produce the nth scanned beam when superposed with the induced

4.0 PAT'rERN SYNTHESIS COMPENSATION TECHNIQUE 129
currents from the other feeds. Superposition is performed with the induced currents of

the composite electronic element excitations.

The amplitudes of the weights a_o are computed assuming the correction patterns

G,,/(u, u_#) are those of a smooth reflector. An extensive scanning study (see Sec. 6.1)

showed that for reflectors with moderate surface errors, the main beam peak

characteristics did not deviate significantly from those of a smooth reflector. These

small deviations in the peak characteristics cause small errors in the _0 calculations but,

in general, do not significantly affect the compensation.

The side lobes of distorted reflector antennas are typically higher than those of

smooth reflectors (see Sec. 2.1). The vast majority of effort with the pattern synthesis

compensation technique deals with side lobe reduction. For side lobe reduction, the

correction beam must satisfy the anti-phase condition with Ft,-,)(u_o ) for correction at

_o. Therefore, the +_180 ° term from (4. l-18) should be used when computing the _bo.

Pattern phase data are also. required for the ISM applied to reflectors. In

implementing the pattern synthesis compensation technique, phase data will probably

not be available. This problem is overcome with an additional step involving an

"amplitude-only" phasor analysis. The auxiliary beam is amplitude weighted using

(4.1-16) and steered to the location of the Ft'-')(u_ '_) pattern requiring compensation. A

blind guess is made at to the required phase excitation of a_o (a good start is to assume

the phase excitation is 0°). By drawing a phasor diagram of the peak of the side lobe

before the auxiliary beam is scanned, the peak after scanning, and the (approximately)

known amplitude of the G,,/(u, u_'_) pattern, the relative phase, _,, between the peak of

the side lobe before compensation and the scanned correction beam can be determined

to a reasonable degree of accuracy (see Fig. 4.2-1). The relative phase is ambiguous as

it could lead or lag the phase of the side lobe. This means that qb, from (4.1-18) is

4.0 PA'I'FERN SYNTHESIS COMPENSATION TECttNIQUE 130
<

Side lobe amp

Correction pat.- ...............

Resultant

/

\

Figure 4.2-1. Phasor analysis to determine the relative phase th, for compensation.

The circle represents the locus of possible values for the correction beam

peak. The amplitude measurements lead to an ambiguous phase

determination in which the correction beam phase could lead or lag the

phase of the side lobe.

4.0 PATTERN SYNTHESIS COMPENSATION TECHNIQUE 131
+ _,. Therefore, at least one extra step (possibly two) is required to resolve the

ambiguity and to set the phase excitation to the proper value for compensation.

The excitation g'_ is computed without any knowledge of the surface error.

Assuming a smooth reflector when calculating the peak of G,,,(u, _ ) causes a slight

error in the phasor determination of the phase _,. This error reduces the accuracy of the

match at the iterative sample point u_O. The error is not catastrophic, however, as there

is a range of phase values over which reduced compensation is achieved.

Figure 4.2-2 illustrates the range of phases for which reduced or full compensation

is achieved when

Fa(u_ 0 ) = 0.316 [ F(t-')(u (0)[ (4.2-i)

(a 10 dB reduction in the side lobe). The side lobe peak is assumed to be normalized to

have unity amplitude with phase 0 ° and is given by

fs,. = l (4.2-2)

Assuming arbitrary phase, the amplitude of the correction beam peak is (from (4.1-16))

a_ =-0.684. The locus of possible values for the weighted correction beam peak is a

circle in the complex plane and is given by

f¢_, = x + jy (4.2-3)

where Ix +jy I = 0.684 or

x 2 +y2 = (0.684)2 (4.2-4)

For a reduction in the side lobe peak, the magnitude of the sum of the peak of the side

lobe peak (4.2-2) and the correction pattern (4.2-3) must be less than unity. Otherwise,

4.0 PATTERN SYNTHESIS COMPENSATION TECHNIQUE 132
/

/

f

-42

\

\

f

I

j,v-axis

2

-- _ 1 fcP

iI x-axis

/

j

-2

Figure 4.2-2. Graph representing the range of phases /'or which reduced or full

compensation is achieved when trying to reduce the side lobe peak by l0

dB. fsc is the side lobe amplitude and fcp is the correction pattern locus.

The range of possible excitation phases for achieving reduced or lull

compensation is represented by the intersection offcp and the interior of

the dashed circle 180 ° -Oo _ [_cp[ < 180° relative to the phase ot'.fsc.

4.0 PA'VrERN SYNTHESIS COMPENSATION TECHNIQUE 133
destructive interference results and the distorted pattern side lobe peak increases.

is represented by the inequality

This

I(x+ 1) +jyl _ 1 (4.2-5)

or by its magnitude squared

(1 + x) 2 +y2 < 1 (4.2-6)

Equation (4.2-6) represents the interior of the dashed circle in Fig. 4.2-2 centered at

(-I,0) with a radius of unity. The inequality of (4.2-6) must hold for reduced or full

compensation. Solving for the points of intersection of (4.2-4) and the interior of the

circle of (4.2-6) gives a range of possible phases for which reduced or full compensation

is achieved. The actual phase of the weighted correction beam peak must fall in the

range

180" - 00 < ]_ct, I < 180" (4.2-7)

relative to the phase offsL. For the above example, 00 = 70*.

4.0 PATI'EI_N SYNTHESIS COMPENSATION TECHNIQUE 134
5.0 COMPUTER CODES

New FORTRAN computer codes and modified forms of existing codes were needed

for the compensation analysis. The programs RAPCA (Reflector Analysis Program for

Cylindrical Antennas), SCANRAP (SCANned Reflector Analysis Program), and

DRAPCA (Dual Reflector Analysis Program for Cylindrical Reflector Antennas) were

written to facilitate the two-dimensional analysis of cylindrical reflectors. The program

RAP (Reflector Analysis Program [52, 53]) was used to analyze paraboloidal reflectors.

RAP was modified slightly to allow rapid evaluation of distorted reflector antennas.

5.0 COMPUTER CODES 135
5.1 REFLECTOR ANAL YSIS PROGRAM FOR

CYLINDRICAL REFLECTORS (RAPCA )

5.1.1 RAPCA ALGORITHM

RAPCA is used to evaluate the radiation patterns for smooth or distorted

cylindrical reflectors. The pattern is computed using physical optics. The reflectors can

have a single feed clement or an array of feed elements. The reflector surface can either

be smooth or distorted. Axisymmctric or offset reflectors with the feed elements tilted

toward the reflector are possible.

Figure 5.1-1 shows the geometry of a cylindrical reflector. The feeds arc assumed

to be longitudinally polarized line sources parallel to the y-axis. The xz-planc is

therefore the H-plane. Following [47], the feeds arc assumed to have radiation patterns

that are described by normalized feed radiation patterns G(F,). For this analysis the

normalized feed radiation pattern is dcf'med to be G(y3 - cos,y_. The feeds are assumed

to have uniform distribution of intensity in the y-direction.

The radiation pattern for the cylindrical reflector in this study with N feed array

elements positioned as shown in Fig. 5.1-1 is given by [23]

'l Ate'/#' 2

P(o)

G_O)= N - Pc/ (s.I-l)

E.4',

5.0 COMPUTER CODES 136
X

D/2

D/2

2_

So

[th feed

-._ fFigure $.1-1. Geometry of the cylindrical reflector used for the program RAPCA.

The reflector is uniform in the y-direction.

5.0 COMPUTER CODES 137
where A, and _b, are the excitation amplitude and phase of the ith feed element, I, is the

radiation integral of the ith element (to be derived below), P(0) is the unnormalized

power pattern of the reflector, and Pt, is the on-axis level one would obtain from a

uniformly illuminated aperture with no spillover. The expression for the total reflector

system (secondary pattern), GN(O), is computed through superposition of the reflector

(secondary) radiation patterns of the individual weighted feed (primary) elements.

The radiation integrals for the individual feed elements can be found from the prime

focus cylindrical reflector analysis in [47]. The H-plane radiation pattern is the only one

evaluated. The analysis in this study is a special two-dimensional case in which the

reflector is assumed to be uniform in the y-direction and the surface errors are also

cylindrical. Integration in the y-direction yields a constant that can be normalized out.

The H-plane radiation pattern is the only one evaluated. The unnormalized radiation

integral for the Rh feed element after integrating in the y-direction is given by (from

(5-86) in 1471)

[ G(y,) ]@ cos fl exp[ -jk(p,- -po R,)] ds It = J" Pl (5.1-2)

where So is the surface of the reflector and ds is the element of arc length along the cross

section of the reflector. This integral can be reduced to an integral in x for the analysis

in this study in which only the xz-plane pattern is to be computed. To see this, the terms

in the integrand need to rewritten in terms of x.

First, we examine the cos p term. This can be found by taking the dot product

- g = p,Igl cosp (5.1-3)

where

5.0 COMPUTER CODES 138
_,= (x - x,)9 + (f(x) - zl)9 (5.1-4)

1

p_ffi I_,1 = [(x- x,): + (z,-fCx))2] T (5.1-5)

N =-f'(x)_ + 2

(5.1-6)

INI =._/l'.f'C._)]2+ l

(5.1-7)

and f(x) is the function that describes the reflector surface, f'(x) is the derivative of

f(x), and z, is the distance from the z ffi0 plane to the ith feed element. Factoring (5.1-3)

and performing the dot product using (5. I-5) and (5.1-6),

cos # ffi 1_ E(x- x_)f'(x) + z_-f(x)3 (5.1-8)

p,INI

in which the cosine is now in terms of functions of x.

Now we examine the phase terms in the exponent of (5.1-2). Define

A

A = Pt-/9 • R_ (5.1-9)

^

The unit vector R, in the xz-plane is given by

/_1 = sin 0._ + cos 0] (5.1-10)

Evaluating the dot product in (5.1-9),

A = Pl - x sin 0 -f(x) cos 0

which is also in terms of functions of x.

The angle Ylis

(5.l-ll)

5.0 COMPUTER CODES 139
x-x,

(5.1-12)

where =t is the tilt angle of the feed elements.

All the terms in the integrand have been rewritten in terms of x. Ev.erything is a

constant with respect to y and the y-integration may be performed giving a constant that

will be accounted for when the radiation integral is normalized. The single integral that

is left is in terms ofds. This may be rewritten in terms ofdx by introducing the Jacobian

in the xz-plane. The relation is

as= x/Ef'(x)32+ 1 ax= l_lax . (S.l-13)

The limits of integration are {x,#-D/2, x,#+ DI2} . Using (5.1-5), (5.1-8), (5.1-11),

(S. 1-12), and (5.1-13) in (5.1-2), the radiation integral for the ith feed element is

Xo#+D/2

;,=cI

x,#-DI2

[ -T, G(Yl) ]'T[/(x) - zj- (x- xl)f'(x)]

x exp[ -jk{p I - x sin 0 -f(x) cos O} ] dx

(5.1-14)

where the constant C is a normalization constant. From [23],

c=J_ (S.l-_S) 2riD '

Equation (5.1-14) is the radiation integral evaluated by RAPCA. It has the

arbitrary function f(x) defining the reflector surface. For the special case of an

axisymmetric parabolic cylinder reflector,

X 2

f(x)= 4-_" (5.1-16)

5.0 COMPUTER CODES 140
where F is the focal length of the parabolic cylinder and Xog--- 0cz = 0 . Using this in

(5.1-14) gives [231

__°,2 2riD _ ]+[2--F- 4-'-F 2]- z,

-D/2

x _l_lpaexp -jk{p l - x sin 0 __ -_X 2 cos 0} ] dx

(5.1-t7)

The function describing the reflector surface used in the analysis of this study was

chosen to be a distorted parabolic cylinder reflector. The particular surface distortion

used sets up a surface error corresponding to a sinusoidal phase error in the equivalent

PO aperture plane current assuming the feed is at the focal point.

For a smooth reflector with the feed at the focus (see Fig. 5.1-1 for the geometry),

d, the sum of the distance from the focal point to a point on the reflector surface and

the distance from the point on the reflector to the aperture plane (containing the focal

point), is a constant 2F (F is the focal length). The equation for d is

d = po +F-f (x) (5.1-18)

where P0 is the distance from the focal point to a point on the reflector

oo= _/x 2 + (F-f (x))2 (5.1-19)

f(x) is the function defining the reflector surface, and d = 2F for the smooth parabolic

reflector. For the distorted surface, a sinusoidal variation in d as a function of x is

required to implement the desired sinusoidal phase error in the equivalent PO aperture

plane current. The expression for d for a distorted reflector is

5.0 COMPUTER CODES 141
d-- 2F+ _ = F-f (x) + Po (5.1-20)

where

= I" cos( 4nmx (5.1-21)

where F is the amplitude of the distortion, m is the number of periods along the radius

of the aperture, and D is the diameter of the aperture. Substituting (5.1-19) into (5.1-20)

and solving forf(x) gives

x 2 - 2F_ - _2

f(x) -- 4F +2_ (5,1-22)

This surface error produces a phase error in the projected aperture plane equivalent

current, OE, given by

(1)E ----_c COS( 41tmx (5.1-23)

where _, - (2n/,l)F. The periodic nature ofthe surface error creates a region ofhigh side

lobes over an angular location determined by the spatial frequency (m) of the surface

error (similar to the radial error from Sec. 2.1.2). [9]

5.1.2 RAPCA INPUT AND OUTPUT

The program RAPCA requires two input Fries which are denoted by the logical file

numbers (LFN) 20 and 21. LFN 20 contains feed and reflector input data and defines

the type of Gauss-Legendre quadrature to be used for the radiation integral. The

5.0 COMPUTER CODES 142
reflector may be smooth or distorted with the surface distortion defined by (5.1-18). The

reflector may be offset and have up to l0 feeds with arbitrary locations and tilt angles.

LFN 21 contains the weights and zeroes for the Gaussian integration. The output file

is designated by LFN 10. Table 5.1-1 shows the form for LFN 20 and LFN 21. Table

5.1-2 contains descriptions of the input variables for LFN 20. Table 5.1-3 shows the

format for inputting the weights and zeros of the Gaussian integration. Appendix A. 1

gives an example of a cylindrical reflector analyzed by RAPCA.

5.2 SCANNED REFLECTOR ANAL YSIS PROGRAM

(SCANRAP)

5.2.1 SCANRAP ALGORITHM

The program SCANRAP uses the method of conjugate field match (CFM, Sec.

2.3.2) to find the excitations of the feed array elements necessary to scan the reflector

to a given angle 0g. SCANRAP computes the received focal region fields of the array

feed elements for a plane wave incident from the desired angle of scan, Os . The

excitations for scanning the reflector to Os are the complex conjugates of the received

focal region fields at the locations of the feed elements. [33] The reflector is assumed to

be smooth when computing the excitations. These excitations for a smooth reflector are

then used with the distorted reflector to scan a correction beam to the region in the

radiation pattern requiring compensation.

5.0 COMPUTER CODES 143
u_

,,-_ t'4

C3_

[-. [-,

¢"4

--* ¢"4

¢e [.:..[.z..

NN

Z

t...

e,;

<

XX

f..f.

o]

_XX

Z<<

C_

Z

"C_

Z

C_

_TJ

"_<

i¢',1

[..,--

m u

N _

,,-I¢",1

Z

N

C_

0

Z

v

©

,v

N

5.0 COMPUTER CODES 144
Table 5.1-2.

RAPCA input variable description for LFN 20.

ARGUMENT

NFEEDS

AEXCIT,

PEXCIT,

XFEED(i)

ZFEED(i)

TFEED(i)

Q

PERRMX

PERRCY

DIAM

OFFSET

FOCAL

THETAB

THETAE

THETAI

NORDER

TYPE*

I

DESCRIPTION

Number of feeds:

Initially limited to 10 or less

R

R

Excitation amplitude for

ith feed (linear)

Excitation phase for

ith feed (degrees)

R

R

X-coordinate of ith feed (i)

Z-coordinate of ith feed (A)

R Tilt of the ith feed (°)

R Feed pattern exponent (cosQ0)

R

R

Phase error amplitude (°)

Number of cycles for the phase error

R

R

R

Reflector diameter().)

Reflector offset dimension (fi)

Focal length of the reflector (2)

R

R

R

Far-field beginning angle (0)

Far-field ending angle (0)

Far-field increment angle (°)

Order of Gaussian quadrature

I: FORTRAN INTEGER

R: FORTRAN REAL NUMBER

5.0 COMPUTER CODES 145
Table 5.1-3.

RAPCA input variable format for LFN 21. Only the positive half of Gauss quadrature

zeros and their corresponding weights are input. RAPCA uses symmetry to fill the ZEROS and WEIGHT arrays. The elements are real variables.

positive, largest root--*

positive, smallest root--*

ZERO(1) WEIGHT(I)

ZERO(2) WEIGHT(2)

ZERO(M) V_EIGHT(M)

(M -- NORDER/2)

5.0 COMPUTER CODES 146
Figure 5.2-1 shows the geometry for the SCANRAP analysis. A linearly polarized

plane wave is assumed to be incident on the cylindrical reflector with the E-field

y-polarized. The expressions for the incident fields are

g= &_-lk_ (5.2-,)

- &

H,= --¢-e-lk'( cos 0s; - sin 0s_)= -T- _ -jk,_, (5.2-2)

where _ is the magnitude of the E-field and 0, is the angle of incidence of the plane wave

and also the angle of desired scan. The scattered E-field from the reflector is given by

[471

(5.2-3)

where So is the surface of the reflector,

-Jkp

@ = e (5.2-4)

J

for the parabolic cylinder reflector, p being the distance from the elemental current on

the reflector to the point of evaluation, and [47]

(5.2-5)

With the given incident fields of(5.2-1) and (5.2-2), J only has a y-directed component

and does not depend on y. It can be shown that for the infinite cylindrical reflector of

Fig. 5.2-1 with a y-polarized feed

7. vCV_O) = 0 . (5.2-6)

5.0 COMPUTER CODES 147
Z

Figure 5.2-1. Geometry of'the cylindrical reflector used for the program SCANRAP.

The reflector is assumed to be a smooth parabolic cylinder that is uniform

in the y-direction.

$.0 COMPUTER CODES

148
The expression for the scattered E-field in (5.2-3) reduces to

e,- k2 LjC, ds (5.2-7)

The received field at the ith feed location is found using the infinitesimal scattered

E-field. [47] The infinitesimal field intensity from (5.2-7) is given by

k 2

dE,- 4rtjoJe JC/dS (5.2-8)

(Note: the vector notation has been dropped as the E field only has a y-component. In

addition, the feed is also y-polarized so there is no need to account for any polarization

mismatch loss.) The magnitude of the Poynting vector of the incremental field intensity

at the ith feed is [47]

1 _ )+ 2 IS, I =T(T laE, I (5.2-9)

The incremental received power of the ith feed is then [47]

dP_= IS, IGI(O,¢).-_-n (5.2-10)

where, for the infinite cylindrical reflector, the feed radiation pattern is defined as

G:(O,4,)= c(y_) (5.2-11)

and is assumed constant in the y-direction. The incremental received voltage of the

ith feed is proportional to the square root of the received power given by

dVt-- a d_ r (5.2-12)

$.0 COMPUTER CODES 149
where a isa constant of proportionality.Using (5.2-8),(5.2-9),(5.2-I0),and (5.2-II)in

(5.2-12),the incremental receivedvoltageis

av,-- (5.2-13)

where all constants are absorbed by the constant C. The constants are not important

to this analysis as the final results will be normalized. The total received voltage for the

ith feed is found by integrating (5.2-13) along the entire surface of the parabolic cylinder

reflector. The integral is

(5.2-14)

where

aS=dyas (5.2-15)

and ds is the element of arc length along the cross section of the reflector. The y-integral

from (5.2-14) yields a constant for this cylindrical analysis (also absorbed into C). The

integral is expressed in terms of dx instead of ds by using the Jacobian. The surface

function of a smooth parabolic cylinder is given by (5.1-16) and is repeated here

X 2

f(x)= 4"_" (5.2-16)

where F is the focal length. Using (5.2-16) in (5.1-13), ds can be rewritten as

ds-- [1 + (-_-F)2] _ dx (5.2-17)

Using (5.2-17), the integral of (5.2-14) becomes

5.0 COMPUTER CODES 150
1/24.0FFSET

1

I[2-OFFSET

The excitation for the ith feed to scan the reflector to angle 0s is then, using (5.2-18),

Aze/#'= V; (5.2-19)

where A, and ¢), are the amplitude and phase of the excitation, respectively, and the star

denotes the complex conjugate (Note: this is essentially (3) from [23] with a constant

weighting function).

This algorithm for computing the excitations follows the DCFM approach (see Sec.

2.3.2). The received focal region fields were computed directly. The complex conjugates

of the received fields were were then output as the array feed element excitations.

SCANRAP computes the excitations using (5.2-19). The excitation with the largest value

of A, is normalized to unity and is used as a phase reference of zero degrees. The other

excitations are then scaled to this normalized maximum excitation.

To evaluate the integral of (5.2-18), all the quantities of the integrand must be in

terms ofx. The expression for y, in terms ofx is given in (5.1-12). The function ¢, in

(5.2-4) is a function ofp which is a function ofx as shown in (5.1-5).

The induced current J can also be expressed as a function ofx using (5.2-5). The

geometry of Fig. 5.2-2 is used to find the expression for .1. The intersection of the x-z

axes is assumed to be the phase reference. The plane wave is incident from angle 0,.

Variables corresponding to the ray passing through the origin are denoted by the

"0"-subscripts. The variables corresponding to the ray passing through the point being

evaluated by the integral are unsubscripted.

Using the intersection of the xz-axes as the phase reference, the H-field of (5.2-2)

can be rewritten With an absolute phase term as

$.0 COMPUTER CODES 151
X ×2/4F

x

(xo-x)=(z -zo)cotO s

xo=zotanE)

5

×0

Z ZO

I1==..=

IlW="-

Z

Figure 5.2-2. Geometry used to Fred Ar in terms ofx.

5.0 COMPUTER CODES

152
-- _ cos OsJ - sin Os2 )

(5.2-20)

The Ar-term needs to be written in terms of x. The two lines defined by

Xo -- Zo tan 0 s (5.2-21)

(x0 - x) = (z - z0) cot Os (5.2-22)

and solving (5.2-22) for x0,

xo --- (z - zo) cot 0 s + x (5.2-23)

will be used to find an appropriate expression for Ar. Substituting (5.2-23) into (5.2-21)

and factoring gives

z cot 0 s + x

z° = tan 0 s + cot 0 s (5.2-24)

Using Fig. 5.2-2, the path length difference Ar is

z o z cot 8 s + x

Ar - - (5.2-25)

cos a s cos as[ tan a s + cot as]

which can be simplified using trigonometric definitions for the tangent and cotangent.

The expression in (5.2-25) becomes

Ar = x sin 8 s + z cos as (5.2-26)

Therefore, substituting (5.2-16) into (5.2-26) gives the desired result for the phase term

of the incident H-field of

5.0 COMPUTER CODES 153
(x) 2

Ar = x sin 0 s + _ cos 0s .

(5.2-27)

The final step in expressing J in terms of x is to evaluate the cross product of

(5.2-5). Using (5.1-6) and (5.1-7), the surface normal is

A X "1

+ _.] (5.2-28) " ,/, +

Using (5.2-28) and (5.2-2), the cross product of(5.2-5) is

,,×h,= _/t+:_z__ -ST _+_ 2F )

X

cos 0 s - --_ sin 0 s

x 2

A

Y

x [" cos Os._ - sin Os_.]

(5.2-29)

If the amplitude of the incident H-field is normalized to unity, the current density, using

(5.2-20), (5.2-27), and (5.2-29) in (5.2-5), is [231

X 2

= 2) COSO S- X _cosOs] -_--f sin 0 s e/kixsmOs+ (5.2-30)

x )2

The current expression in (5.2-30) is used in (5.2-18) and the required current excitation

for the ith feed is then found with (5.2-19)

5.0 COMPUTER CODES 154
5.2.2 SCANRAP INPUT AND OUTPUT

The program SCANRAP requires two input files which are denoted by the logical

fde numbers (LFN) 20 and 21. LFN 20 contains feed and reflector input data and

def'mes the type of Gauss-Legendre quadrature to be used for the computation of the

feed element excitations. The reflector is assumed to be smooth and can be offset. The

feed can have up to 10 elements with arbitrary locations and tilt angles. LFN 21

contains the weights and zeroes for the Gaussian integration. It is the same as LFN 21

used by the program RAPCA. The output file is designated by LFN 10. Table 5.2-1

shows the form for LFN 20 and LFN 21. Table 5.2-2 contains descriptions of the input

variables for LFN 20. Refer to Table 5.1-3 for the format of LFN 21. Appendix A.2

gives an example of computing the excitations of an array feed for a given scan angle.

5.3 DUAL REFLECTOR ANAL YSIS PROGRAM FOR

CYLINDRICAL REFLECTORS (DRAPCA)

5.3.1 DRAPCA ALGORITHM

DRAPCA is used to evaluate the radiation patterns for dual cylindrical reflector

antennas. The radiation pattern is computed using geometrical theory of diffraction

(GTD) analysis for the scattered field from the subreflector and physical optics/aperture

integration (PO/AI) for the main reflector. The dual reflector can have multiple feeds.

5.0 COMPUTER CODES 155
Table5.2-1.

SCANRAP input file formats for LFN 20 and LFN 21.

LFN 20

NFEEDS

XFEED(I)

XFEED(2)

XFEEI3(NFEEDS)

Q

DIAM

THETAS

NORDER

ZFEED(I) TFEED(I)

ZFEED(2) TFEED(2)

ZFEEI)(NFEEDS) TFEED(NFEEDS)

OFFSET FOCAL

LFN 21

ZEROS(I)

ZEROS(2)

ZEROS(NORDER)

WEIGHT(1)

WEIGHT(2)

WEIGI_T(NORDER)

5.0 COMPUTER CODES

156
Table 5.2-2.

SCANRAP input variable description for LFN 20.

ARGUMENT

NFEEDS

XFEED(i)

ZFEED(i)

TFEED(i)

Q

DIAM

OFFSET

FOCAL

THETAS

NORDER

TYPE*

I

R

R

R

R

R

R

R

R

I

DESCRIPTION

Number of feeds:

Initially limited to 10 or less

X-coordinate of ith feed (2)

Z-coordinate of ith feed (3.)

Tilt of the ith feed (°)

Feed pattern exponent (cos_t0)

Diameter of the aperture (3.)

Aperture offset dimension (2)

Focal length of the reflector (,t)

Scan angle (°)

Order of Gaussian quadrature

I: FORTRAN INTEGER

R: FORTRAN REAL NUMBER

5.0 COMPUTER CODES 157
The subreflector surface is assumed to be a hyperbola (Cassegrain) or an ellipse

(Gregorian). The main reflector is a parabola and can either be smooth or distorted.

The distorted main reflector surface is the same as described in Sec. 5.1.1, equations

(5. l-18), (5. l-19), and (5.1-20). Axisymmetric, single offset, or dual offset reflectors with

tilted feed elements are possible.

Figure 5.3-1 shows the geometry of a dual cylindrical reflector. The feeds are

assumed to be linearly polarized line sources parallel to the y-axis. As in Section 5. l, the

feeds are assumed to have radiation patterns described by G(y,)= cos_),,. Also, as in

Section 5.1, the radiation pattern for a dual cylindrical reflector with N feed array

elements is given by (equation (5.1-1) is repeated here for convenience) [23]

N

lm l

G,v(O) = N (5.3-1)

where A, and _b, are the excitation amplitude and phase of the ith feed element and I, is

the radiation integral of the ith element (to be derived below). The total radiation

pattern, G_O), is the superposition of the weighted radiation patterns of the individual

feeds. (Actually, the superposition is performed in DRAPCA with the induced currents

on the main reflector as this is computationally much more efficient).

There are two coordinate systems used to evaluate the dual cylindrical reflector (see

Fig. 5.3-1). The radiation pattern coordinate system is centered at F0 with the positive

x-axis pointing up and the positive z-axis pointing to the right. The feed coordinate

system is centered at FI and it is located by the tilt of the interfocal axis (fl) and the

leftmost loci of the ellipse or hyperbola. The positive z-axis of the feed coordinate

system is along the ray from F1 to the center of the subreflector.

5.0 COMPUTER CODES 158
DMAIN/2

t

DMA_N/2

Figure 5.3--1. Dual cylindrical reflector geometry.. This is the geometry DRAPCA

uses to analyze the two-dimensional problem.

5.0 COMPUTER CODES 159
DERIVATION OF THE UNNORMALIZED RADIATION INTEGRAL

The radiation integrals for the individual feed elements can be found by extending

the prime focus cylindrical reflector analysis in [47]. The analysis for this program is a

special case in which the reflectors are assumed uniform in the y-direction. Integration

in the y-direction yields a constant that can be normalized out. The remainder of the

unnormalized radiation integral for the ith feed is (following [47])

so

where ,_ is the induced PO current on the main reflector and So is the surface ofthe main

reflector. The integral along the profile of the main reflector surface, ds , can be

replaced by an integration in x by making the substitution from (5.1-13). The integral

can be rewritten as

OFFM+DMA IN/2

[_----_ _e Jk(-{''_') INIdx (5.3-3)

OFFM-DMAINI2

The PO current J, has three components: a GTD component from the top edge of

the subreflector (-It) , a GTD component from the bottom edge of the subreflector

(is), and a GO (specular) component from the subreflector surface (is) The total PO

current is then

(5.3-4)

The PO current is related to the H-field incident upon the main reflector. The PO

current for the ith feed is given by 2h ×/_,, for m- T,B,S. Each component of the

5.0 COMPUTER CODES 160
H-field can be expressed in terms of the incident E-field as (constants will be normalized

out)

A A A

oc rm x Etm = rm x y Etm (5.3-5)

where E,,, is

component is

the complex E-field amplitude. The induced PO current for the ruth

L o__ x r,. A xyEl,,, A (5.3-6)

The cross product can be simplified. Referring to Fig. 5.3-2, the cross product in (5.3-6)

is

x rm x y Eun = E#. sin_ = Ev. cos ZY (5.3-7)

but

E_ cosx= - Et_(;_•_)= - E_ (5.3-8)

Equation (5.3-8) can be substituted into (5.3-4) to give

:, = (- Err_r- E_B_B- E_s_s)"_ (5.3-9)

which when used in (5.3-3) produces the desired form

OFFM4-DMA INI2

I,= I

OFFM-DMA INJ2

(5.3-10)

which is the unnormalized radiation integral for the ith feed evaluated by DRAPCA.

5.0 COMPUTER CODF.S 161
rmx Y

Y

Z

Figure 5.3-2. Vectors used to simplify the cross product expression of(5.3-7).

5.0 COMPUTER CODES

162
E-FIELD EXPRESSIONS FOR PO CURRENT CALCULATIONS

The problem remains to find expressions for the E-fields incident upon the main

reflector. The GTD fields from the top and bottom edges of the subreflector are of the

same form. The GO field from the subreflector surface has a distinct form.

The GTD E-field incident upon the main reflector is found by analyzing each ofthe

subreflector edges as if it were an infinite half-plane. The incident E-field of the ith feed

due to one of the edges is given by (using [55])

1 -j_, (5.3-11)

Eo,, -- - DIIEi_ _r m e

where m-- T or B, D u is the uniform theory of diffraction (UTD) coefficient and E,r,_is

the field from the ith feed incident upon the subreflector edge. The E,_ is defined as (any

constants will be normalized out)

E_= G(Y_m) Te-Jm'"r= c°sqYtm Te-Jkr"'r

rim F rim F

(5.3-12)

where y,,, is the ith feed angle in the direction of the top or bottom edge. Substituting

(5.3-12) in (5.3-11),

cosqYlm ]+ 1 +,,_,F) (5.3-13) E_,,, - - DII rlmr _ e -jkO''

The UTD diffraction coefficients are found using the subroutine in Appendix G.9, [55].

The GO E-field incident upon the main reflector is determined using ray optics.

The GO E-field component corresponds to the ray reflected from the specular point on

the subreflector. The cylindrical GO E-field incident upon the main reflector is [551

5.0 COMPUTER CODES 163
/31 T e-J_s (5.3-14)

where Er_ is the field incident upon the subreflector as defined in (5.3-12), p_ is the radius

of curvature of the reflected wavefront, and the minus sign accounts for the reversal of

the E-field vector upon reflection from the subreflector. Using(5.3-12) in (5.3-14) gives

the desired formulation of the GO component

E ]' ]' cosqyts Te-j_,s F Pl Te-l_ s

Ets-- - r_se P l + rs

(5.3-15)

The radius of curvature of the reflected wavefront is computed using [55]

I I 2

+ (5.3-16) _Igg

Pl rtsF Pc cos 0'

where po is the radius of curvature of the subreflector at the specular point and O' is the

angle of incidence or reflection at the specular point.

5.3.2 DRAPCA INPUT AND OUTPUT

The program DRAPCA requires two input files which are denoted by the logical

file numbers (LFN) 20 and 21. LFN 20 contains feed and reflector input data and

def'mes the type of Gauss-Legendre quadrature to be used for the radiation integral. The

main reflector may be smooth or distorted with the surface distortion defined by (5.1-18).

The dual reflector may be offset or dual offset. The subreflector may be hyperbolic or

elliptical. There may be up to 10 feed elements with arbitrary locations and tilt angles.

LFN 21 contains the weights and zeroes for the Gaussian integration. It is the same as

5.0 COMPUTER CODES 164
LFN 21usedby the program RAPCA. The output file is designatedby LFN 10. Table

5.3-1showsthe form for LFN 20 and LFN 21. Table 5.3-2containsdescriptionsofthe

input variablesfor LFN 20. Referto Table 5.1-3for the format of LFN 21. Appendix

A.3 gives an example ofa Cassegrain reflector analyzed by DRAPCA.

5.4 REFLECTOR ANAL YSIS PROGRAM (RAP),

VERSION 2.0

5.4.1 RAP ALGORITHM

RAP is used to evaluate the radiation patterns for prime focus paraboloidal

reflector antennas. The radiation pattern is computed using PO/AI. The reflector

surface can be smooth or distorted. The projected aperture can have arbitrary shape.

The reflector may be offset. Feeds displaced from the focus and tilted are possible.

References [461 and [531 describe the algorithm in detail.

RAP, 2.0, has a surface interpolation routine that can be used to evaluate distorted

or shaped reflectors. Interpolation, however, is computationally time consuming.

Evaluation of the analytical distortion is much faster than for an interpolated surface.

The RAP subroutine REFLEC which analyzes the surface function was modified slightly

to evaluate distorted reflectors with analytical expressions. The equations defining the

surface and derivatives of the surface were changed to allow pattern computation for

distorted reflectors.

5.0 COMPUTER CODES 165
I

¢"4

Z

e_

¢.,4

Z

•=_ ,_

,,.=,

<

[... [.

.=-, ¢"4

NN

•-. C'4

_X

f_

c_

Z

cL

[- O

Z

N O

[._

N

_N N

ZZ<<

•_ z

=• 0

<(Y_rm_Z

,,=,IN

M

m

iJ ----

O_

0

Z

° .

r_

Z

0

5.0 COMPUTER CODES

166
ARGUMENT

NGTD

NFEEDS

ISYS

AEXCIT,

PEXCIT,

XFEED(i)

ZFEED(i)

TFEED(i)

Q

PERRMX

PERRCY

DMAIN

OFFM

FOCAL

DSUB

ECC

TWOC

OFFS

OFFF

THETAB

THETAE

THETAI

NORDER

Table 5.3-2.

DRAPCA input variable description for LFN 20.

TYPE*

I

DESCRIPTION

Controls the subreflector analysis:

NGTD = 0 - GTD is used

NGTD-- 1 - GO is used

Number of feeds:

Initially limited to 10 or less

Defines the feed input location:

ISYS = 1 - FI coordinates

ISYS = 2 - Primed coordinates

R

R

Excitation amplitude for

ith feed (linear)

Excitation phase for

ith feed (degrees)

R

R

X-coordinate of ith feed (2)

Z-coordinate of ith feed (2)

R Tilt of the ith feed (°)

R

R

R

Feed pattern exponent (cosQ0)

Phase error amplitude (°)

Number of cycles for the phase error

R

R

R

Main reflector diameter (2)

Main reflector offset (2)

Focal length, main reflector (i)

R

R

R

R

R

Subreflector diameter ().)

Subreflector eccentricity

lnterfocal distance (2)

Subreflector offset (2)

Feed phase center offset (t)

R

R

R

Far-field beginning angle (°)

Far-field ending angle (°)

Far-field increment angle (°)

Order of Gaussian quadrature

h FORTRAN INTEGER

R: FORTRAN REAL NUMBER

5.0 COMPUTER CODES 167
Two types of distorted surfaces were evaluated. Both had periodic errors. Both

produce regions of high side lobes with a level dependent upon the amplitude of the

errors; the location of the high side lobes depends on the frequency of the error across

the surface.

The distorted surface function is found in a similar manner to the distorted surface

function used by ILAPCA (see Sec. 5.1.1, (5.1-22)). Figure 5.4-1 gives the reflector

system geometry. For this geometry, the distance d is

d_r'-z' (5.4-1)

where

r' = _/( a' )2+ (z')5 (5.4-2)

z' defines the surface function, and for a smooth parabolic reflector, d = 2F. Following

the derivation of the cylindrical error in See. 5.1.1, the distorted surface is defined to be

d-_ 2F + _ = r' - z' + _ . (5.4-3)

Substituting (5.4-2) into (5.4-3) and solving for z', the distorted reflector surface

expression is

p2 _ (2F + _):

z' =f( p, p,, ¢o) = 4F + 2_ (5.4-4)

where p, p., and qb, are the cylindrical coordinates defining the surface (see Fig. 5.4-2),

F is the focal length, and _ is the function describing the surface error. This reduces to

the equation of a smooth paraboloid for the no error case of _ --- 0.

5.0 COMPUTER CODES 168
x(x')

l \

\

8

I I

I I

I

/

y(yl)

Z$ X$

Figure 5.4-1. Geometry def'ming paraboloidal reflector used in RAP. [52]

5.0 COMPUTER CODES

169
D/2

D/2

P

a Y

OFFSET

Focal point

Figure 5.4-2. Geometry def'ming the radial and azimuthal surface errors.

5.0 COMPUTER CODES

170
The first paraboloidal surface distortion case analyzed was periodic as a function

of radius. The distortion gives rise to a sinusoidal error in the aperture equivalent

current. The radial surface error is given by

- F cos( 41rmPa D ) (5.4-5)

where F is the magnitude of the surface error (in _), m is the number of periods along

the radius, and D is the diameter of the aperture. The phase error in the aperture

equivalent current is

@E = eg cos( 4rtmpa 0 )(5.4-6)

where _b_ = (2nl_l)F.

The second paraboloidal surface distortion case analyzed contained periodic errors

in aperture azimuth angle _b,. This distortion gives rise to a scalloped phase error that

somewhat approximates the error when a radial rib reflector is used. The azimuthal

surface error is given by

2

(= r(y-¢os I ¢os,.¢.1) . (s.4-7)

This error has zero mean. The corresponding phase error in the aperture equivalent

current is

2

_e= ee(_- cos I cosine°l) (5.4-8)

The profiles of both surface distortions are shown in" Fig. 5.4-3.

5.0 COMPUTER CODES 171
(]a} a

Figure 5.4-3. Surface error prof'fles for the (a) radial error and the (b) azimuthal

error.

5.0 COMPUTER CODES

172
5.4.2 RAP INPUT AND OUTPUT

References [52] and [53] describe the RAP input and output files in detail. Example

programs are given in the references.

5.0 COMPUTER CODES

173
6.0 COMPENSATION RESULTS

Results of studies using both cylindrical and paraboloidal reflectors are presented

in this chapter. An extensive scan profile was generated for the cylindrical reflector to

evaluate the scanned beams of smooth and distorted reflectors. Compensation results

are presented for both cylindrical and paraboloidal reflectors, using displaced auxiliary

feeds and fLxed array feeds that use individual element amplitude and phase control

(electronic compensation) to scan the compensation beams.

6.0 COMPENSATION RESULTS 174
6.1 SCANNING STUDIES

6.1.1 PRIME FOCUS CYLINDRICAL REFLECTOR SCANNING

STUDY

The pattern synthesis technique correction beams are meant to cause localized

constructive effects but the side lobe envelope degrades with increasing scan. This

degradation eventually produces a destructive global effect on the radiation pattern.

Therefore, the side lobe degradation will ultimately limit the angular range over which

the pattern synthesis compensation technique will be effective. For this reason, scanning

studies were performed on cylindrical reflectors. Displaced feed scanning (prime focus,

Sec. 3.1) and electronic scanning (See. 3.2) were evaluated. The scanning studies were

used as a guideline for determining the limits over which the compensation technique

can be used. In addition, the studies were used to determine whether the amplitude and

phase of the scanned beam were significantly different between the smooth and distorted

reflector cases.

For the scanning studies, and all the cylindrical prime focus reflector studies, the

feed pattern was assumed to be a cos,0 pattern with q = 3. This gives a -12 dB edge

taper with F/D = 0.4. The surface distortion of(5.1-21) and (5.1-22) is used. It sets up

a surface error corresponding to a sinusoidal phase error in the equivalent PO aperture

plane current.

6.0 COMPENSATION RESULTS 175
FEED DISPLACEMENT

Examples of unscanned smooth and distorted reflector patterns computed using

RAPCA are shown in Fig. 6.l-1. Note that the pattern angle where the region of high

side lobes occurs increases as the frequency of the phase error increases (m increases).

Also, the amplitude of the side lobes increases with increasing amplitude of the phase

error.

Scanning of the beam causes the side lobe envelope to rise and the gain to drop.

Figures 6.1-2 and 6.1-3 show examples of scanned smooth and distorted reflector

radiation patterns. Note that while the high side lobe regions caused by the surface

distortions appear in nearly the same location relative to the main beam, the main beam

is nearly unaffected by the surface distortions. The scanned pattern in Fig. 6.1-3,

however, is not usable for compensation as the side lobes are too high. Compensation

for this reflector would then be limited to a range inside the location of the scanned

beam shown in Fig. 6.1-3.

It is necessary to have amplitude and phase information for the scanned beam in

order to apply the pattern synthesis compensation technique (see Sec. 4.2). The

amplitudes and phases of the scanned beams produced by lateral displacement of the

feed for the smooth and distorted reflector mentioned above were compared (see Table

6.1.1). The important point to note from the scan study is that, while the side lobe

regions of the scanned patterns (e. g., Figs. 6.1-2 and 6.1-3) were noticeably affected by

the surface distortion, the characteristics of the peak of the scanned beams for the

distorted reflector did not vary greatly from the smooth reflector. This means that the

amplitudes and phases of the auxiliary beams that will be used for compensation are

essentially known based on smooth reflector performance. This is a major result because

it implies that knowledge of the actual surface deviations is not requh'ed. It is also

6.0 COMPENSATION RESULTS 176
-i0

-4O

-50

\

t

0.0 1.0 2.0

-t0

-40

l

t.O 2.0 3.0 4.0 S.O 8.0 7.0 8.0 g.O tO.O

AmL[ [_'W

(b)

Figure 6.1-1. Unscanned beam for (a) smooth reflector, (b) distorted reflector with

2 periods of the surface error along the radius, and (c) distorted reflector

with 5 periods of the surface error along the radius. The surface errors are

such that the sinusoidal phase error of (5.1-23) appears across the

aperture with a peak phase error of bE= 20 ° The

F/D = 0.4, D = 100,1.

6.0 COMPENSATION RESULTS 177
-tO

-4O

-5O

i

-60 , ,

0.0 t.0 2.0 3.O a.O 5.O S.O 7.O e.O 9.0 tO.O

_mGLE_6)

(c)

Figure 6.1-1. (continued)

6.0 COMPENSATION RESULTS 178
W

I.-

I,-4

.-I

-t0

-2O

-3O

-4O

-50

-60

0.0 t.O _.0 3.0 4.0 S.O 5.0 7.0 e.o 9.o to.o

_U_LE_S)

Figure 6,1-2, Scanned reflector patterns for lateral feed displacement of 23.. The

cases are for smooth reflector (solid) and distorted reflector (dashed) with

2 periods of the surface error along the radius (F/D = 0.4).

6.0 COMPENSATION RESULTS 179
-to

-40

-5o

-6O

0.0 i.o e.o 3.0 4.0 5.0 6.0 7.0 a.o 9.0 io.o

(DES)

Figure 6.1-3. Scanned reflector patterns for lateral feed displacement of 4,t. The

cases are for smooth reflector (solid) and distorted reflector (dashed) with

2 periods of the surface error along the radius (F/D = 0.4). This beam is

beyond the useful limit of our compensation technique as the side lobes

are too high.

6.0 COMPENSATION RESULTS 180
,2.'

_o

f.,) II

_._

_._

L.

U _

V

,,4 _.

,-4

¢',1 ,_

e4

esu. ii _

e't._ ,.-(',4

_m

-- .2_' _' -- -&

_ N _ _ m N

0 c;

_ II U II U U

6.0 COMPENSATION RESULTS 181
possible to determine the relative phase of the side lobe by using amplitude only

measurements. This phasor analysis is presented in Section 6.2. I.

A presentation by Lee [21] at the NASA Earth Science Geostationary Platform

Workshop held at NASA Langley Research Center prompted investigation into

extending the scan range by using a higher F/D ratio reflector. Figure 6. I-4 shows

scanned beams for a cylindrical reflector with F/D-- 1.0, D-- I002, and q -- 3 . The

higher F/D reflector has significantly less increase in the side lobe envelope for a given

scan angle than the lower F/D reflector evaluated above. This led to an investigation

into extending the useful range of the pattern synthesis compensation technique by using

a higher F/D reflector (see Section 6.2.1). The higher F/D reflector, however, would

require a larger supporting structure and, if electronic scanning is used, a larger feed

array. This, in turn, motivated investigation into a dual reflector configuration (see Sec.

6.4).

ELECTRONIC SCANNING

The cylindrical scanning study concluded with examination of electronic scanning.

Again, a 1002 cylindrical reflector with F/D= 0.4 was used. The feed was a 7-element

linear array with 0.52 element spacing, centered on the focal line, and orthogonal to it.

The element pattern was the same as the single element in the above laterally displaced

feed analysis ( cos,0, q = 3).

Figure 6.1-5 shows examples of electronically scanned beams. The excitations of

the elements were computed with SCANRAP (Sec. 5.2). Table 6.1-2 gives the element

excitations for the electronic scans. Comparison to Figs. 6.1-2 and 6.1-3 reveals that the

electronically scanned beams show improved performance compared to the single,

6.0 COMPENSATION RESULTS 182
\

\

\

%

Figure 6.1-4. Radiation patterns for a smooth 100). reflector with F/D = 1.0 for the

cases of unscanned (solid), lateral feed displacement of 52 (dashed), and

lateral feed displacement of 10_. (dot-dash).

6.0 COMPENSATION RESULTS 183
i • i 1

-lO

-30

-40

-50

V '

i

/ .i "\\

: /

I *

!i " I

\

\

\

\

,. \

t , • ,

_'_ /.

, ¢" .

• : ,' /

: : , o ,

--_ | * I

-4.0 -3.0 -2.0 -t.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0

Figure 6.1-5. Electronically scanned beams for a cylindrical reflector with F/D = 0.4

and D-- 100,L A 7-element array feed with 0.5_ -element spacing was

used. The element excitations were computed with SCANRAP and are

given in Table 6.1-2.

6.0 COMPENSATION RESULTS 184
Table 6.1-2.

Feed excitations for the scanned beams of Fig. 6.1-5.

the focal plane (Z-COORD = 40,t).

The feed elements are located in

FEED EXCITATION FOR 0 DEGREE SCAN

FEED X-COORD AMPLITUDE

(,_) (LINEAR)

PHASE (DEG)

1 1.5000 .0900 .6204

2 1.0000 .1089 179.6889

3 .5000 .1457 -.8344

4 .0000 1.0000 .0000

5 -.5000 .1457 -.8344

6 -l.0000 .1089 179.6889

7 -1.5000 .0900 .6204

FEED EXCITATION FOR 1 DEGREE SCAN

FEED X-COORD AMPLITUDE

().) (LINEAR)

PHASE (DEG)

1 1.5000 .0465 ,3289

2 1.0000 .0586 179.1039

3 .5000 .0709 -1.4059

4 .0000 .0700 178.9586

5 -.5000 .7045 -.4076

6 -1.0000 1.0000 ,0000

7 -1.5000 .2934 -178,2593

FEED EXCITATION FOR 2 DEGREE SCAN

FEED X-COORD AMPLITUDE

(),) (LINEAR)

PHASE (DEG)

I 1.5000 ,0122 -.8269

2 1.0000 .0212 175.9401

3 .5000 .0298 -5.4128

4 .0000 .0373 174.1605

5 -.5000 .0547 -5.3645

6 -I.0000 .2113 -.8740

7 -1.5000 1.0000 .0000

6.0 COMPENSATION RESULTS 185
laterally displaced feed. The array feed is able to somewhat compensate for the inherent

phase errors associated with scanning a prime focus reflector by capturing more of the

energy that is spread in the focal region due to scanning and adding it coherently. With

the above 32-diameter feed array configuration, the reflector is limited to about ±3

beamwidths of scan. This limk could be increased by using a larger array.

As with the scanned beams using laterally displaced feeds, it was found that the

characteristics of the scanned beam peaks for the distorted reflector using a phased array

feed did not vary greatly from the smooth reflector. This means that for the electronic

scanned case that the amplitudes and phases of the auxiliary beams used for

compensation are essentially known.

6.1.2 DUAL REFLECTOR SCANNING STUDIES

The dual reflector scanning study was performed on an infinite cylindrical

axisymmetric Cassegrain reflector (see Fig. 6.1-6). The main reflector had a diameter

of D = 1002 with F/D = 0.5. The subreflector had a diameter of 342 with eccentricity

e = 3. The focal length of the equivalent prime focus reflector is [191

Fe= F M = 50 M (6.1-I)

where [191

e+l

M= -2 (6.1-2) e-I

6.0 COMPENSATION RESULTS 186
F/D=50/_ O0

DSUB=34-, e=3

feed

Z

Figure 6.1-6. Infinite cylindrical axisymmetric Cassegrain reflector.

prime focus reflector has F_/D = 1.0.

The equivalent

6.0 COMPENSATION RESULTS 187
so F_ --- I00, The choices of main reflector and subreflector eccentricity then give rise to

an equivalent reflector with Fe/D -=-1.0. Blockage was neglected. Feeds displaced from

the focal point were used to scan the reflectors.

For small angle scans, the Cassegrain exhibits similar characteristics to that of its

equivalent prime focus reflector (see Sec. 3.3). The equivalent reflector in this case has

F_/D m 1.0 An example of a pattern for the Cassegrain with the feed laterally displaced

by 22 in the minus x-direction is shown in Fig. 6.1-7. Note the phase is reasonably fiat

across the main beam. For wide angle scans, however, the phase aberrations of the

Cassegrain differ significantly from the equivalent prime focus reflector. [I 3] This is now

discussed further.

A quantitative comparison of wide angle dual reflector and prime focus reflector

scanning is shown in Fig. 6. I-8, Fig, 6. l-8(a) shows the pattern for the Cassegrain with

a feed laterally displaced -9.£_ in the minus x=direction to scan the reflector to 5.7 °. Fig.

6.1-8(b) shows the pattern for the equivalent prime focus reflector (a single cylindrical

reflector, F/D = 1.0) with the feed laterally displaced -I0,_ in the minus x-direction to

scan the reflector to 5.7 °, Note that the pattern phase across the main beam of the

Cassegrain has a very steep slope while the phase across the main beam of the equivalent

parabola is fairly flat. The steep ramp of the phase will make it dimcult for the pattern

synthesis compensation technique to coherently match the distorted pattern phase over

a significant portion of the correction beam. In addition, the main beam of the

Cassegrain pattern in Fig. 6, l=8(a) is very broad which impairs the ability of the pattern

synthesis technique to act locaUy on a distorted pattern.

Krichevsky and DiFonzo [15] reported on optimizing the offset Cassegrain

scanning performance by displacing the feed laterally and away from the focal plane (see

Sec. 3.3). The optimized patterns had considerably better scanned pattern characteristics

than those for lateral displacement in the focal plane. [15] The optimum loci for the

6.0 COMPENSATION RESULTS 188
0-1

W

V

w -20

o'3

<

-r--30

SCANNED

DRAPCA

I

\

A \

I

I

I

I

q

REFLECTOR

\

I

I

I

?

-- AMP

PHASE

N

\

I

I

I

I

8O

44

O8

72

36

_0

_-40 72

<_-50 1 O8

iI iI

( ,, ! v V] 1.

-60 _ 1 80

0 1 2 3 4. 5 6 7 8 9 10

ANGLE (DEG)

Figure 6.1-7. Scanned reflector pattern for lateral feed displacement of 2t for an

infinite cylindrical Cassegrain reflector. Both the amplitude (solid) and

phase (dashed) are shown.

6.0 COMPENSATION RESULTS 189
_'-10

hJ

_'-20

Oq

<

-r"

0.-30

nn

E3-40

v

n

-50

-60

0

I

k

II

Ii

II

I II '

I I I

t I I

I I t

I I I

1 II I

\ _,1 \

I. , 1 . =1, .

1 2 3

SCANNED REFLECTOR

OR,u=,C,,,, [__ AUp i ; PHASE

t r _ L I t 180

A/ X', ;,, ,.

11 I \/ _ I _ I I 108

It I I

" rV i\,, ,, ,,

' " I '.-I_O,_ I 1

/ ', ! 111-I_1 0

.,' ,--i;

/ '-I 11 11 'i'.I -72

I . _ i i j 11 is-I - 108

I/ '! It'1 11, !I_144

4 5 6 7 8 9 10

ANGLE (DEC)

(a)

-12

-18

W

-24 ,<

-r

-_0

Q.

• --36

--42

--48

--54

SCANNED PATrERNS

RAPCA OUTPUT

• _ I"" !

I f_ f_ f_ ft_

,I 80

144

1 00

72

36

0

-36

-72

-1 08

-1 44

-180

0

j I

• . a . • . • • • • | • | .

-60 .....

0 1 2 3 4 5 6 7 8 9

ANGLE

(b)

Figu_ 6.1-8. Amplitude (solid) and phase (dashed) for scanned reflector patterns

(scan angle is 5.7°). (a) Dual cylindrical Cassegrain with the feed laterally

displaced. (b) Equivalent prime focus reflector. (c) Dual reflector with

the feed optimally displaced.

6.0 COMPENSATION RESULTS 190
0 \

\

_'-I0

LIJ I

E3 I

V

L,J -20 t

O3 t

< I

"1- I

la_ m -30

-40 I

-50 I!i 'I

I

I I iI

-60 -' ' :

0 1

SCANNED REFLECTOR

DRAPCA

I ,_ Ill \I ",\

,_ A!/ v --\

k,,V v, ! \

_,v, , ',

I l \

II l

II I II

II l II

II I II

II l_ I

l I t , , I • , • • i • I L

2 3 4. 5 6 7

ANGLE (DEG)

\ I\

\ I\

I I I

I I l

I I I

1 80

1 44

1 08

72

36

(_kl^ I I 10

li t-1 O8 -1 44

..... -1 80

8 9 10

(c)

Figure 6.1-8. (continued)

6.0 COMPENSATION RESULTS 191
feeds are along a hyperbola that curves towards the subreflector as the feed is displaced

farther from the focal point. [15] Following the offset dual reflector optimum feed

location analysis discussed in [15], trial and error was used to ['md a near optimum feed

location for the wide angle scan. The feed was displaced by -9.52 in the minus

x-direction and 4.52 in the plus z-direction. Fig. 6.1-8(c) shows the pattern for the

optimized feed location. The slope of the phase across the main beam is much less

severe and the main beam is much narrower.

The wide angle scanned pattern of Fig. 6.1-8(c) can be used for pattern

compensation with the pattern synthesis technique. The dual configuration

demonstrates improved scanning of the higher virtual F/D dual reflector over that of a

lower F/D prime focus reflector while maintaining a compact feed structure.

6.2 CYLINDRICAL REFLECTOR COMPENSATION

A NA L YSIS

6.2.1 COIVlPENSATION WITH AN AUXILIARY FEED

Compensation calculations were run for the F/D=402/lO0,). distorted reflector

corresponding to the pattern in Fig. 6.1-1(b). An auxiliary scanned beam produced by

a second feed was used to reduce the peak of the high side lobe caused by the surface

distortion. The element pattern of the auxiliary feed was assumed to be the same as the

feed at the focal point (cos,0, q = 3) and it was aimed along the boresight direction.

The excitations a_o were computed using (4.1-19). To illustrate the pattern synthesis

6.0 COMPENSATION RESULTS 192
compensation technique with auxiliary feeds, it is useful to view the process from a

superposed pattern standpoint. From (4.1-12), the resultant pattern after the single

iteration is

FO)(u) = F (°) + AFO)(u) (6.2-1)

where F _ is the original distorted pattern and the correction pattern with one feed is

(6.2-2)

where the a_1_is the weight of the correction beam.

The plots in Fig. 6.2-1 show two cases of compensation. In Fig. 6.2-1(a), the

auxiliary scanned beams were weighted to produce nulls at the locations of the peaks

of the high side lobes. The auxiliary feeds were displaced by -t-1.882 to scan the

correction beams to the locations of the high side lobes. To produce nulls, the F_(_ '_)

from (4.1-16) would be equal to zero. The weights were computed assuming a smooth

reflector to be consistent with actual implementation of the technique. Using the

distorted reflector data from Fig. 6.1-1(b) and the assumed smooth reflector scanned

pattern data, the amplitude and phase of the correction beam excitations were computed

using (4.1-16) and (4.1-18). The feed element excitations for the compensated pattern

in Fig. 6.2-1(a) are found in Table 6.2-1. This weighting would produce absolute nulls

at the locations of the peaks of the high side lobes if the weights were computed with the

scanned correction beams of the actual distorted reflector. Even without knowledge the

surface distortion, the pattern synthesis compensation technique produced the pattern

of Fig. 6.2-l(a) with significant reduction in the high side lobes of the distorted reflector

pattern of Fig. 6.1-1(b).

Figure 6.2-1(b) shows a case where the amplitude of the weights were reduced to

produce a 10 dB reduction in the side lobe peaks. Once again, the weights were

6.0 COMPENSATION RESULTS 193
-lO

-t.0 0.0 1.0

_SLE_

(a)

2.0 3.0 4.0 5.0

°lO

-40

-50

-6O

/ :

P.,' I

'.] '..

! I

I !

/

f _

'_1_ ° 0 -4.0 -3.0 -_°0 "_.0 0.0 1,0

.°,

i

i

F\!__

,,_ ,,

ii a

t i t

i i i

i i i

i i I

i

., I ',

2.O 3.O

Iy

4.0 5.0

(b)

Figure 6.2-1. Compensation For the high side lobes at +_2.3 ° orthe distorted reflector

in Fig. 6.1-1(b) using auxiliary scanned beams generated by lateral

displacement or two feeds (F/D=0.4). (a) The auxiliary beams are

weighted to produce nulls. (b) The auxiliary beams are weighted to

reduce the side lobe peaks by 10 dB.

6.0 COMPENSATION RESULTS 194
Table 6.2-1.

Feed excitations for the displaced feed compensated patterns in Fig. 6.2-1. The elements

are located in the focal plane (Z-COORD = 40,t). The two cases are ['or (a) placing nulls

at the peaks of the high side lobes and (b) reducing the peaks of the high side lobes by

10 dB.

FEED EXCITATION FOR NULL

FEED X-COORD AMPLITUDE PHASE (DEG)

(A) (LINEAR)

1 1.0000 .0000 .0000

2 .1960 o1.8800 99.5300

3 .1960 1.8800 99.5300

(a)

FEED EXCITATION FOR 10 dB REDUCTION

FEED X-COORD AMPLITUDE PHASE (DEG)

(2) (LINEAR)

1 1.0000 .0000 .0000

2 .1340 °1.8800 99.5300

3 .1340 1.8800 99.5300

(b)

6.0 COMPENSATION RESULTS 195
computed assuming a smooth reflector. To reduce the peak by 10 dB, the F_(u_O) from

(4.1-16) would be equal to 0.316F¢'-,(u_O). The feed locations and excitations for the

compensated pattern in Fig. 6.2-1(b) are given in Table 6.2-1. The advantage to the

reduced amplitude case is that the lower auxiliary beam amplitude causes less

interference with the surrounding side lobe envelope. This is important in keeping the

effects of the pattern synthesis compensation technique iterations localized.

EXTENDED RANGE COMPENSATION

Compensation analysis was run on a distorted cylindrical reflector with an

F/Dr 100MI00L The improved side lobe envelope of the scanned beams for the higher

F/D-ratio reflector allows compensation to be performed much farther out than the

lower F/D-ratio case. Figure 6.2-2 shows a compensation example illustrating the

extended range of the higher F/D-ratio reflector; the F/D = 0.,4 case was limited to about

+3" compensation range. The correction beam is weighted to reduce the side lobe peak

by 10 dB.

SIDE LOBE PHASE DETERMINATION

Figure 6.2-1 was generated assuming that the pattern phase was known. In

implementing the pattern synthesis compensation technique, phase data will probably

not be available. This problem is overcome with the "amplitude-only" phasor analysis

discussed in Sec. 4.2. The auxiliary beam is amplitude weighted as above and steered to

the location of the pattern requiring compensation with a blind guess as to the required

phase excitation (a good start is to assume the phase excitation is 0°). By drawing a

6.0 COMPENSATION RESULTS 196
-|0

-4O

--50

o

-t0

-20

-30

-40

-50

|

\

I

t

g

1.0

| ! a I | e e |

_ |

2.0 3.0 4.0 5,0 S.O 9.0

kNSLE_6}

(a)

7.0

1 | i , i i i

0.0

g

t0.0

i

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 _0.0

kNGLE (:DES)

(b)

Figure 6.2-2. Radiation patterns for a distorted 100_. reflector with a distortion

having five spatial periods along the radius, F/D = 1.0 for the cases of (a)

uncompensated and (b) compensated. A second displaced feed was used

to generate an auxiliary correction beam positioned at 5.76 °.

6.0 COMPENSATION RESULTS 197
phasor diagram of the peak of the side lobe before the auxiliary beam is scanned, the

peak alter scanning, and the (approximately) known amplitude of the auxiliary beam,-

the relative phase between the side lobe before compensation and the scanned beam can

be determined to a reasonable degree of accuracy by determining the angle _, from Fig.

4.2-I. The relative phase is ambiguous as it could lead or lag the phase of the side lobe.

Therefore, at least one extra step (possibly two) is required to set the phase excitation

to the proper value for compensation.

The above compensation case in Fig. 6.l-l(b) was used as a start point and

analyzed without knowledge of the distorted pattern phase. The desired result is to

reduce the peak of the high side lobe by l0 dB as was done previously in Fig. 6.2-1(b).

Figure 6.2-3 illustrates the process. In the first iteration, the scanned correction beam

was weighted with the excitation amplitude from Fig. 6.2-1(b) and a phase of 0 °. The

resulting pattern (solid curve in Fig. 6.2-3)) actually caused the side lobe peak to increase

by 0.22dB. After determining the relative phase with the phasor analysis, the

compensated pattern (dashed curve in Fig. 6.2-3)) gave results comparable to Fig.

6.2-1(b). The dot-dash curve in Fig. 6.2-3 is the case where the ambiguity in the relative

phase caused an error in determining the phase.

6.2.2 ELECTRONIC COMPENSATION

In this Section, the pattern synthesis compensation technique is implemented with

electronic scanning of the correction beams. To illustrate the pattern synthesis

compensation technique with electronic scanning, it is useful to view the process from

a superposed current standpoint. Using (4.1-15), the composite induced current on the

reflector due to the ruth feed after this single iteration,

6.0 COMPENSATION RESULTS 198
-i0

-20

I -30

-40

-50

1 i i l :

t.O 2.0 3.0 4.0 5.0

[DES)

-6O

0.0 6.0 7.0 8.0 9.0 t0.0

Figure 6.2-3. Radiation patterns of a distorted _'lindrical reflector, F/D = 40A/100_t,

for compensation without pattern phase data. The phasor analysis

described in See. 4.2 was used. The original distorted pattern is that of"

Fig. 6.1-1(b). The solid curve is the first iteration with auxiliary' beam

phase excitation of 0 °. The dashed curve is the second iteration alter the

relative phase has been determined. The dot-dash curve is the case where

the relative phase ambiguity caused an error in determining the phase.

6.0 COMPENSATION RESULTS 199
frO) ¢(o) (I)

Gf,fft_ Jref,m _ Afref,m (6.2-3)

and, using (4.1-14),

(i) _(I)_(i) (i)O)

Af_ef_ ffi ul _l + a,j gin2 (6.2-4)

where the ai_, a_I_ are the compensation weights for each of the two beams and the

g_2{' g_2]are the induced currents on the reflector for each of the two beams due to the

ruth feed. The composite current of (6.2-3) is then evaluated in the radiation integral to

get the compensated pattern.

Electronic compensation was performed on the distorted reflector corresponding

to the pattern in Fig. 6.2-4. The distorted reflector has a slightly lower surface error

spatial frequency (m) than the reflector from Fig. 6.1-1(b). The feed array was the same

seven element array used previously in Section 6.1. Two auxiliary scanned beams

(symmetrically placed on each side of the main beam) given in Fig. 6.2-5 were

simultaneously used to reduce the high side lobes caused by the surface distortion. The

excitations for the scanned beams were computed with SCANRAP assuming a smooth

surface. The excitations for scanning the beams were then weighted to reduced the side

lobe peaks by 10 dB and superposed to get the compensation excitations using phase

information computed from the distorted pattern. The compensated pattern is shown

in Fig. 6.2-6. The excitations and locations of the array feed are listed in Table 6.2-2.

6.0 COMPENSATION RESULTS 200
A

Figure 6.2-4. Unscanned patterns of an axisymmetric reflector for the cases of a

smooth surface (solid) and a distorted surface (dashed) with 1.5 periods

of the surface error across the reflector. The FID = 0.4 and D-- 100).

The maximum phase error in the PO current is _-- 20 ° .

6.0 COMPENSATION RESULTS 201
0

-t0

-4O

-50

Figure 6.2-5. Auxiliary electronic scanned beam from a linear feed array of seven

A/2-spaced elements used to reduce the high side lobe at 1.72 ° caused by

the surface error across the reflector of Fig. 6.2-4. The pattern was

computed assuming a smooth reflector (solid curve). The actual scanned

pattern of the distorted reflector is also shown (dashed). A similar beam

is used to reduce the high side lobe at -1.72 °.

6.0 COMPENSATION RESULTS 202
-i0

,_ -20

-30

-4O

-5O

-6O

-5.0 -4.0 -3.0 -2.0 -t.O 0.0 t.0 2.0 5.0

ANGLE (DE6]

Figure 6.2-6. The pattern resulting From application or compensation/'or the high

side lobes at +1.72 ° using the auxiliary scanned beams of Fig. 6.2-5

produced by a linear Feed array of'seven )./2-spaced elements (solid). The

dashed line is the uncompensated pattern and the dot-dash line is the

pattern of a smooth reflector. The amplitudes of the weights were

designed to produce a 10 dB reduction in the side lobes.

6.0 COMPENSATION RESULTS 203
Table 6.2-2.

Feed excitations for the electronically compensated patterns in Fig. 6.2-6. The elements

are located in the focal plane (Z-COORD--402). The excitations were computed to reduce the peak of the high side lobe by 10 dB.

FEED EXCITATION FOR 10 dB REDUCTION

FEED X-COORD AMPLITUDE PHASE (DEG)

(2) (LINEAR)

1 1.5000 .1150 100.2700

2 1.0000 .0620 99.5300

3 .5000 .0017 104.9100

4 .0000 .9990 .1976

5 -.5000 .0017 104.9100

6 - 1.0000 .0620 99.5300

7 -1.5000 .1150 100.2700

6.0 COMPENSATION RESULTS 204
6.3 PARABOLOIDAL REFLECTOR

COMPENSATION ANAL YSIS

Compensation calculations were run for distorted prime focus paraboloidal

reflectors with F/D = 0.5 and D-- 100/l. The feed elements are linearly polarized and

have a radiation pattern approximated by a cos00 with q = 2.7 (this q gives an edge

illumination of-12 dB). The process uses auxiliary beams that are scanned to the

locations in the pattern requiring compensation. The aux beams are then weighted to

correct pattern irregularities. The weights are computed assuming a smooth reflector.

The pattern synthesis compensation technique was implemented using displaced feed

scanning and electronic scanning. The distorted reflectors studied had surface errors that

gave rise to periodic aperture plane equivalent current phase errors. The errors were

periodic in either the radial or azimuthal directions. The RAP, Version 2.0, subroutine

REFLEC that defines the reflector surface was modified to allow implementation of a

reflector with surface errors that give rise to the periodic aperture plane current phase

errors. The surface functions for the distorted surfaces are given in Sec. 5.4. Both kinds

of surface errors produce regions of high side lobes in which the amplitude of the side

lobes depends on the amplitude of the phase error and the location depends on the

spatial frequency of the error.

6.0 COMPENSATION RESULTS 205
6.3.1 COMPENSATION WITH AUXILIARY DISPLACED FEEDS

The radial surface error defined by (5.4-4) and (5.4-5)was chosen so that a

sinusoidalerroras a function of radius occurred in the aperture plane equivalentcurrent

phase. The radialerrorcauses a narrow region ofhigh sidelobes. Figure 6.3-I shows

the compensation case where four displacedauxiliaryfeeds(placed symmetrically about

the axis) were used to compensate for high side lobes in the principal planes, The

correction beams were weighted to reduce the side lobe peaks by I0 dB. The feed

element locations and excitations are given in Table 6.3-I. The actual peak reductions

are given in Table 6.3-2.

The azimuthal surface error defined by (5.4-4) and (5.4-7) was chosen so that a

scalloped phase error occurred in the aperture plane current. This error somewhat

approximates the error that occurs when a radial rib-type reflector is used. The

azimuthal surface error causes a distortion over a wider region than the radial surface

error does (comparing the distorted patterns from Figs. 6.3-I and 6.3-2). Figure 6.3-2

shows the compensation case where four auxiliary feeds (placed symmetrically about the

axis) were used to compensate for high side lobes in the principal planes. The

compensation beams were weighted to reduce the side lobe peaks by 10 dB.

Repeated implementation of the pattern synthesis technique was used to further

improve on the E-plane radiation pattern of Fig. 6.3-2a. Two more pairs of auxiliary

beams were scanned to each side of the main beam in two separate steps to reduce the

side lobe envelope by 10 dB at +3.4 ° and _+4.6 °. The results are shown in Fig. 6.3-3.

The feed element locations and excitations are given in Table 6.3-I. The actual peak

reductions are given in Table 6.3-2.

6.0 COMPENSATION RESULTS 206
-10

-40

-So

-I10

, J

f o B

II

°,V

-I 0 -I I -I I -I I -0 1 0 | | l I I I I

l_l.l IOll!

(a)

i." 3.0

-10

i -20

-40

-50

/+

/.........\ /

I:/ it' !i

_/V'.i ii il

-S0

-lO -I I -t l -I I -411 l O III I I l 8 I I l O

llell.lloll!

(b)

Figure 6.3-1. Compensation for radial surface distortions in a paraboloidal reflector

(F/D = 0.5, D = 10Ok)using four feeds displaced symmetrically about the

axis. (a) E-plane and the (b) H-plane. The dot-dash curves are the

undistorted radiation patterns. The dashed curves are the distorted

radiation patterns with a phase error of _E--30° with one period along

the radius. The solid curves arc the compensated patterns using displaced

feeds to scan the compensation beams.

6.0 COMPENSATION RESULTS 207
-I0

-t0

.',,

' t _ I ,%

i , I i * •

I i i e 1 # i f4

o t I _ t e •

i n o |

i

!

, eo* ifet w

i *e I P

ii II i

| Ii I I i i I I

t if e_

ol oJ el /

/

• i f _1

2.0 3.0 4.O 5.0

(b)

Figure 6.3-2. Compensation for azimuthal surface distortionsin a paraboloidal

reflector(F/D = 0.5, D = 10Gl)using four feeds displaced symmetrically

about the ax_s. (a) E-plane and the (b) H-plane. The dashcd curves arc

the undistorted radiationpatterns. The dot-dash curvcs arc thc distorted

radiationpatterns with a phase error or _b_= 60° with 4 periods around

the perimeter. The solid curves are the compensated pattcrns using

displacedfeeds to scan the compensation beams.

6.0 COMPENSATION RESULTS 208
-10

=50

-tO

._ -20

°(o

i o i

i i |

-4.0 -3.0 -2.0 -t.O 0.0 1.0 2.0 3.0 4.0 5.0

J_.O -! .0

p'\

VU: I

-4.0 -3.0 -2.0

:: ,, ,

.,k/-"

) i i ) m

,* ,, ,, ,, , , , ,',

i) )4 *) ) i a _

,i I) ) i i * i t

! i) )) _ ) t i

i t ) ) ,

• :, ,o , :, , ,

I _ . ,' ,, :,

0.0 $.0 2.0 3.0 4.0 5.0

a_SL£ (DES)

(b)

Figure 6,3=3. Further improvement of the E-plane radiation pattern of Fig. 6.3-2(a).

The dashed curves are the undistorted radiation patterns. The dot-dash

curves are the distorted radiation patterns. The solid curves are the

compensated patterns using displaced feeds to scan the compensation

beams. (a) Additional beams are placed at +3.,4 °. (b) Additional beams

are placed at 4-4.6 ° .

6.0 COMPENSATION RESULTS

209
Table 6.3-1.

Feed excitations for paraboloidal compensated patterns. The elements are located in the

focal plane. The excitations were computed to reduce the peaks of the high side lobes

in the principal planes by 10 dB. The two cases are for (a) periodic radial surface errors

(Fig. 6.3-1) and (b) periodic azimuthal surface errors (Fig. 6.3-3(b)).

FEED EXCITATION FOR 10 dB REDUCTION - RADIAL ERRORS

FEED X-COORD Y-COORD AMPLITUDE PHASE (DEG)

(,t) (,!) (LINEAR)

1 .0000 .0000 1.0000 .0000

2 1.4400 .0000 .0705 102.3000

3 - 1.4400 .0000 .0705 102.3000

4 .0000 1.4400 .0705 102.3000

5 .0000 - 1.4400 .0705 102.3000

(a)

FEED EXCITATION FOR 10 dB REDUCTION - AZIMUTHAL ERRORS

FEED X-COORD Y-COORD AMPLITUDE

(_l) (,t) (LINEAR)

PHASE (DEG)

I .0000 .0000 1.0000 .0000

2 2.2000 .0000 .0415 102.3000

3 -2.2000 .0000 .0415 102.3000

4 .0000 2.2000 .0415 102.3000

5 .0000 -2.2000 .0415 102.3000

6 3.4000 .0000 .0180 176.1000

7 -3.4000 .0000 .0180 176.1000

8 4.6000 " .0000 .0090 172.4000

9 -4.6000 .0000 .0090 172.4000

6.0 COMPENSATION RESULTS 210
Table 6.3-2.

Side lobe peak reduction for compensated paraboloidal patterns.

at the location of the correction beam peaks is given in this table.

distorted reflector patterns in Figs. 6.3-1, 6.3-2, 6.3-3, and 6.3-5.

The peak reduction

The data are for the

Figure Peak

Location

6.3-1 +1.4 °

6.3-2 4-2.2 °

6.3-3 +3.4 °

-t-4.6 °

6.3-5 4-1.4 °

SL Peak

Reduction

9.6 dB

8.9 dB

8.9 dB

6.9 dB

8.1 dB

6.0 COMPENSATION RESULTS 211
To illustrate the pattern synthesis technique with the repeated implementation

using auxiliary feeds, it is useful to view the process from a superposed pattern

standpoint. From (4.1-12), the resultant pattern after the three iterations is given by

F(3)(u) .. F (o) + E AF(O(u) .

iml

(6.3-1)

where F_) is the original distorted pattern and the correction patterns for the three

iterations are

4

AFO)(u) = _.. a(n0 Grol(U, u(n0 ) (6.3-2)

nml

which represents the four correction beams steered to the locations in the principal

planes,

2

AFt2)(u) -- _ an(2) G,e/(u, un_) ) (6.3-3)

nml

which represents the two correction beams steered to the +3.4 ° locations in the E-plane,

and

2

AFO)(u) '= _. a_3) Grol(U, u¢,3) ) (6.3-4)

tlml

which represents the two correction beams steered to the -I-4.6 ° locations in the E-plane.

6.0 COMPENSATION RESULTS 212
6.3.2 ELECTRONIC COMPENSATION

Electronic compensation was performed for a distorted paraboloidal reflector with

a radial surface error. The feed was a 37-element triangular grid array feed with

half-wavelength spacing and 7 elements along the diagonal (see Fig. 6.3-4). The existing

computer code RAP, Version 2.0, was used to perform the analysis by repeatedly

running the program for each feed and superposing the patterns. The compensation was

performed in the principal planes by electronically scanning four correction beams to the

desired locations and weighting them to produce a 10 dB reduction in the side lobe

amplitude. The method of conjugate field match (ICFM from Sec. 2.3.2) was used to

compute the feed excitations required to electronically scan reflector beams to the four

locations in the principal planes. RAP was used to compute the far field pattern in the

desired direction of scan for each of the individual feeds assuming a smooth reflector.

The complex conjugates of the far field pattern values were then used as the excitations

for the array feed elements to electronically scan the correction beams (the infinite

cylinder electronic compensation case (Sec. 6.2.2) used SCANRAP (See. 5.2) and DCFM

(See. 2.3.2) to compute the element excitations for the correction beams). Each element

excitation was then weighted for the compensation. The weighted patterns were then

superposed with the original distorted pattern. Figure 6.3-5 shows the results of the

electronic scanning. The compensation beams were weighted to reduce the side lobe

peaks by 10 dB. The actual peak reductions are given in Table 6.3-2.

6.0 COMPENSATION RESULTS 213
Y

A

0.5k

Figure 6.3-4. The geometry for the 37-element triangular grid array feed used to

perform electronic compensation with a paraboloidal reflector.

6.0 COMPENSATION RESULTS 214
-10

-50

--60

-10

-40

-50

• %

/

I

P

t | | t

-3.0

!

-E.4

1 ! l ! | ! o

-i.8 -i.2 _.8 0.0 0.8 I.E f.8

(a)

| ! | | i i

'l i t '

!

a

-2.4

1

2.4 3.0

i 1 1 | i ! | e

-$.0 -t.8 -i.2 -0.6 0.0 0.8 S.2 S..8 2.4 3.0

roLE (ZXEm

(b)

Figure 6.3-5. Electronic compensation for radial surface distortions in a paraboloidal

reflector (F/D = 0.5, D = 100_l)using the 37-element array of Fig. 6.3-4.

(a) E-plane and the (b) H-plane. The dot-clash curves are the undistorted

radiation patterns. The dashed curves are the distorted radiation patterns

with a phase error of 4)t = 60° with 4 periods around the perimeter. The

solid curves are the compensated patterns using displaced feeds to scan

the compensation beams.

6.0 COMPENSATION RESULTS 215
6.4 DUAL REFLECTOR COMPENSATION

A NA L YSIS

As discussed in Sec. 3.1, improved scanning is possible using a reflector with a

higher F/D. The higher F/D reflector would require a larger support structure for the

feed. This motivated investigation into a dual reflector configuration with a higher

virtual F]D to extend the compensation range of the pattern synthesis technique.

In this study, the dual reflector analysis is limited to in_t'mite cylindrical problems

using the computer code DRAPCA (Sec. 5.3). There were no 3-dimensional dual

reflector codes available during this study. The dual reflector is axisymmetric and has

a main reflector that is distorted (F/D -- 0.5, D = 100_l) with a surface error that produces

a sinusoidal phase error in the aperture equivalent current. This is the same type of

surface error used in the prime focus infinite cylindrical reflector analysis (eqns. (5.1-21)

and (5.1-22)). The subreflector is hyperbolic with eccentricity e = 3 and diameter

DSUB-- 342 (see Fig. 6.1-6).

Figures 6.4-1 through 6.4-3 illustrate an extended range compensation of a high

side lobe caused by a distorted reflector. The surface error produces a sinusoidal phase

error in the aperture equivalent current with 5 periods along the radius and an amplitude

_e-- 20 ° . Thisis the same error that was used for the extended range compensation

analysis in Sec. 6.2.1. The distorted pattern is shown in Fig. 6.4-1. The scanned

correction beam from a displaced auxiliary feed is shown in Fig. 6.4-2. The correction

beam is weighted to reduce the high side lobe by 10 dB and the resultant compensated

pattern is shown in Fig. 6.4-3. The feed element locations and excitations are listed in

Table 6.4- I.

6.0 COMPENSATION RESULTS 216
-4O

-50

I I I i I i

q

I

I

I

I

I

I

,., !

l

l

l

I

I

i •

T

I

I

: : = .

i

i I i ! i

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

ANGLE_D£S]

Figure 6.4-1. Unscanned patterns for a dual cylindrical axisymmetric reflector. The

F/D = 0.5 and D = 100). for the main reflector. The subreflector has

e = 3 with DSUB = 34).. The patterns are for a smooth (solid) and a

distorted (dashed) reflector. The distortion gives rise to a sinusoidal phase

error in the aperture plane current with 5 periods along the radius and an

amplitude of _b_ = 20*.

6.0 COMPENSATION RESULTS 217
0

-t0

-20

-40

-5O

I ¢ * | i | i ! i

'-"

| AA/_ ,_ N _ /

VV"'v _,"_ VV_"'_,",'

I i i i I i i I I

0.0 t.O 2.0 3.0 4.0 5.0 8.0 7.0 8.0 9.0

_U_SLE_S)

tO.O

Figure 6.4-2. Scanned patterns for the dual cylindrical axisymmetric reflector. The

patterns are for a smooth (solid) and a distorted (dashed) reflector. The

scanned patterns are caused by a displaced auxiliary feed. The feed was

displaced by 9,_ in the minus x-direction and 4.5_ towards the

subreflector. The feed was tilted up l0 °

6.0 COMPENSATION RESULTS 218
-i0

-40

-50

I

I

I

I

. I

',i _,; !f '

I I! II

I I |I II

I I II II

i| t

II

I

-60 ..........

0.0 t.O 2.0 3.0 4.0 5.0 §.0 7.0 8.0 9.0

k_L[ (DES)

I !

I

iO.O

Figure 6.4-3. Compensation for the distorted reflector of. The auxiliary beam of Fig.

6.4-2 was weighted to produce a 10 dB reduction in the side lobe peak.

6.0 COMPENSATION RESULTS 219
Table 6.4-1.

Feed excitations for the displaced feed compensated pattern in Fig. 6.4-3.

excitations were computed to reduce the peak of the high side lobe by 10 dB.

The

FEED EXCITATION FOR 10 dB REDUCTION

FEED X-COORD Z-COORD

(_) (_)

TILT AMPLITUDE

(DEG) (LINEAR)

PHASE (DEG)

I .0000 26.0000 .0000 1.0000 .0000

2 -9.0000 30.5000 10.0000 .1340 133.4400

6.0 COMPENSATION RESULTS 220
The auxiliary feed used for this wide angle compensation was displaced in the

x-direction but also had to be moved towards the subreflector (out of the focal plane)

and tilted (as opposed to the focal plane displacement for the prime focus reflector with

no feed tilt) to achieve an acceptable scanned pattern. This optimized location for the

feed was found by trial and error (see Fig. 6.1-8(c)) but follows the general trends in feed

locations used to optimize offset dual reflectors. [15, 16]

6.0 COMPENSATION RESULTS 221
7.0 SUMMARY AND CONCLUSIONS

The pattern synthesis approach was shown to be a feasible method for

electromagnetically compensating for surface errors in large reflectors. The technique

was developed using cylindrical reflectors and was extended to paraboloidal reflectors.

In this summary we highlight points of note.

Characteristics of previous approaches to electromagnetic compensation:

* They require an accurate model of the surface of the distorted reflector.

• The techniques produce global pattern effects.

• The algorithms are a somewhat blind approach to correcting the patterns (no

pattern data).

The pattern synthesis compensation technique is an alternative to those approaches.

Characteristics of the pattern synthesis technique:

7.0 SUMMARY AND CONCLUSIONS 222
It does not require knowledge of the surface.

It does need radiation pattern data. If surface data are available, however, the

pattern data that are required can be computed.

The pattern synthesis approach uses a localized algorithm in which pattern

corrections are directed specifically towards the portions of the pattern requiring

improvement. "Customizing" is possible by iterating.

The scan results from Section 6.1 provided important results concerning the auxiliary

beams that are used to provide compensation. Results of the scanning study:

The analysis showed that the scanned beam amplitude and phase of a moderately

distorted reflector did not differ significantly from that of the equivalent smooth

reflector. This was shown in Table 6.1-1. Therefore, the amplitude and phase of the

scanned auxiliary beam is essentially a known quantity.

The plots provided insight into estimating the angular range over which the pattern

synthesis compensation technique is effective (Figs. 6.1-2 and 6.1-3). The limitation

is determined by the degradation in the side lobe envelope of the scanned beam.

Scan range may be increased by using a high F[D reflector (Fig. 6.1-4).

however, requires a larger feed displacement than with low F/D reflectors.

This,

The dual reflector scanning analysis showed that for wide angle scanning, a

non-planar feed displacement might be required due to the steep phase ramp across

the main beam (Fig. 6.1-8).

7.0 SUMMARY AND CONCLUSIONS 223
The compensation results of Sections 6.2, 6.3, and 6.4 showed that:

The pattern synthesis compensation technique can be used to improve irregularities

in the radiation patterns of distorted reflectors without knowledge of the surface

distortion.

* The pattern synthesistechnique can be implemented by using auxiliarydisplaced

feeds or a fLxedarray feed.

The method can be extended to wider angle scans by using a higher F/D reflector

(Fig. 6.2-2) or a dual reflector with a higher virtual F/D (Fig. 6.4-3).

The method for determining the phase of the high side lobes using amplitude

measurements only provides a method for overcoming the problem of deciding how

to phase the excitation of the auxiliary beam (Fig. 6.2-3).

Recommendations for future efforts to continue the work of this compensation

study would be to :

Implement the method with 3-dimensional dual reflectors.

Apply the pattern synthesis compensation technique to improve scanned pattern

characteristics.

Investigate hybrid compensation techniques where, for example, CFM would be

used as a Rrst iteration to improve the pattern of a distorted reflector. The pattern

synthesis technique could be used to further shape the compensated pattern.

7.0 SUMMARY AND CONCLUSIONS 224
Develop scenarios for obtaining the necessary pattern data.

The pattern synthesis compensation technique was shown by simulation to be a

feasible method for pattern correction of the effects of surface errors. Experimental

implementation needs to be performed to validate the theory.

7.0 SUMMARY AND CONCLUSIONS 225
8.0 REFERENCES

Ii

*

)

*

*

.

*

.

R. J. Acosta, "Compensation of Reflector Surface Distortions Using Conjugate

Field Matching," 1986 IEEE AP/S Symposium, Philadelphia, PA, June, 1986.

R. Acosta, A. Zaman, E. Bobinsky, A. R. Cherrette, S. W. Lee, "Case Study

of Active Array Feed Compensation with Sidelobe Control for Reflector

Surface Distortion," 1988 AP/S International Symposium, Syracuse, New York,

June, 1988.

R. N. Assaly, L. J. Ricardi, "A Theoretical Study of a Multi-Element Scanning

Feed System for a Parabolic Cylinder," IEEE Trans. on Ant. and Prop., vol.

AP-14, no. 5, pp. 601-605, September, 1966.

S. J. Blank, W. A. Imbriale, "Array Feed Synthesis for correction of Reflector

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9. APPENDIX A: EXAMPLE COMPUTER

INPUTS AND OUTPUTS

A.I R,4PC,4 INPUT AND OUTPUT

The Reflector Analysis Program for Cylindrical Antennas (RAPCA) was discussed

in Sec. 5.1. In this section, an example is presented to illustrate the use of the program.

The example is that of a smooth axisymmetric reflector with diameter D = 1002 and

F[D=0.4 The feed is a seven element linear array with 2/2 -spacing located

symmetrically about the focus in the focal plane. The feed elements each have a cos_0

field pattern with q = 3.0.

It is desired to electronically scan the reflector to Os = 1.5" . The excitations for

scanning the reflector were computed with the program SCANRAP (see Sec. A.2). The

input file for the analysis is shown in Table A.I-1.

quadrature zeros and weights is shown in Table A.1-2.

analysis is shown in Table A. 1-3.

The file containing the Gauss

The output file for the RAPCA

9. APPENDIX A: EXAMPLE COMPUTER INPUTS AND OUTPUTS 231
Table A. 1-1.

RAPCA input file for an axisYmmetric reflector with a seven element linear array feed.

Refer to Tabs. 5.1-1 and 5.1-2 for the format and variable descriptions. The element

excitations were computed using SCANRAP (see Sect. A.2). The direction of scan is

Os = 1.5" .

7

.30003E-01 179.8680 1.5000

.41161E-OI -1.8880 1.0000

•51191E-01 177.2723 .5000

.59228E-01 -2.8996 .0000

.56011E-01 -.5240 -.5000

.10000E+01 .0000 - 1.0000

.83784E+00 .3778 -1.5000

3.0000 0.00 0.00

100.0000 0.0000 40.0000

-5.0 5.0 0.1

2O

40.0000 0.0000

40.0000 0.0000

40.0000 0.0000

40.0000 0.0000

40.0000 0.0000

40.0000 0.0000

40.0000 0.0000

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 232
Table A. 1-2.

Input file containing the Gauss quadrature zeros and weights for order 20.

is outlined in Tab. 5.1-3.

The format

0.9931285992

0.9639719273

0.9122344283

0.8391169718

0.7463319065

0.6360536807

0.5108670020

0.3737060887

0.2277858511

0.0765265211

0.0176140071

0.0406014298

0.0626720483

0.0832767416

0.1019301198

0.1181945320

0.1316886384

0.1420961093

0.1491729865

0.1527533871

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 233
Table A. 1-3.

RAPCA output file for an axisymmetric reflector with a seven element linear array feed.

The direction of scan is Os-- 1.5 °.

RAPCA, VERSION 2.0, SEPTEMBER 12, 1988

**********FEED DATA**********

THERE ARE 7 FEEDS

THEIR EXCITATIONS (LINEAR AND DEGREES)

AND LOCATIONS (WAVELENGTHS) ARE:

FEED AMP PHASE X-COORD Z-COORD

1 .30003E.01 179.8680 1.5000 40.0000

2 .41161E-01 -1.8880 1.0000 40.0000

3 .51191E-01 177.2723 .5000 40.0000

4 .59228E-01 -2.8996 .0000 40.0000

5 .56011E-01 -.5240 -.5000 40.0000

6 .10000E+01 .0002 -1.0000 40.0000

7 .83784E + 00 .3778 - 1.5000 40.0000

TFEED

.0000

.0000

.0000

.0000

.0000

.0000

.0000

THE FEED COSINE EXPONENT = 3.0000

THE MAX PHASE ERROR = .0000 DEGREES

THE NUMBER OF PERIODS OF THE PHASE ERROR = .0000

**********REFLECTOR DATA**********

REFLECTOR DIAMETER (WAVELENGTHS) = 100.0000

REFLECTOR OFFSET (WAVELENGTHS) = .0000

REFLECTOR FOCAL LENGTH (WAVELENGTHS) = 40.0000

REFLECTOR F/DP = .40000

ILLUMINATION AT THE TOP OF THE REFLECTOR (DB) -- -24.7339

ILLUMINATION AT THE BOTTOM OF THE REFLECTOR (DB) = -24.3993

**********OUTPUT PATTERN**********

THETA MIN =

THETA MAX --

THETA INC --

-5.0000

5.0000

.1000

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 234
Table A. 1-3 (cont.).

**********INTEGRATION DATA**********

THE QUADRATURE IS ORDER 20

ZERO WEIGHT

-.99313E+00

-.96397E + 00

-.91223E+00

-.83912E+00

-.74633E + 00

-.63605E + 00

-.5 I087E + 00

-.37371E+00

-.22779E + 00

-.76527E-01

.76527E-01

.22779E + 00

.37371E+00

.51087E + 00

.63605E + 00

.74633E + 00

.83912E+00

.91223E+00

.96397E + 00

.99313E+00

.17614E-01

.40601E-01

.62672E-01

.83277E-01

.10193E+00

.11819E+00

.13169E+00

.14210E+00

.14917E+00

•15275E + 00

•15275E + 00

.14917E+00

.14210E+00

.13169E+00

.ll819E+00

.10193E+00

.83277E-01

.62672E-01

.40601E-01

.17614E-01

THE MAXIMUM PATTERN VALUE IS .41827E+00 AT 1.5000 DEGREES

ANGLE (DEG) MAGNITUDE (DB) PHASE(DEG)

-5.000 -50.77,:12 16 i .,:1400

-4.900 -47.0391 160.1114

-4. 800 -46.5008 159.5966

-4.700 -48.8605 158.7413

-4.600 -57.3711 153.5934

-4.500 -56.8248 - 10.0209

-4.400 -48.1858 - 14.8784

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 235
Table A.I-3 (cont.).

-4.300 -45.5103

-4.200 -45.7109

-4.100 -49.1133

-4.000 -62.7939

-3.900 -52.2489

-3.800 -46.1372

-3.700 -44.4495

-3.600 -45.7089

-3.500 -51.6651

-3.400 -59.1566

-3.300 -46.1309

-3.200 -42.1262

-3.I00 -40.9736

-3.000 -41.9847

-2.900 -45.6230

-2.800 -54.8777

-2.700 -55.6846

-2.600 -50.9732

-2.500 -56.0384

-2.400 -50.4225

-2.300 -40.7853

-2.200 -35.9868

-2.100 -33.4517

-2.000 -32.5962

-1.900 -33.4387

- 1.800 -36.7134

- 1.700 -47.0390

-1.600 -42.9594

-1.500 -35.1966

-1.400 -32.4636

-1.300 -32.1787

-1.200 ,_ -34.2833

- 1.lO0 -40.8624

- 1.000 -48.2606

-.900 -36.0942

-.800 -32.6204

-.700 -32.2052

-.600 -34.7020

-.500 -43.7974

-.400 -40.9616

-.300 -33.0929

-.200 -30.6321

-.lO0 -31.2431

•000 -36.2634

•100 -44.3099

.200 -31.0636

-15.7190

-16.2188

-17.4229

-31.1731

171.4322

168.9309

168.3735

168.0314

166.8671

-5.0962

-9.4387

-I0.0111

-10.5191

-11.5695

-14.6369

-31.8958

-156.5971

-168.5477

-151.7078

-23.2950

-11.2526

-8.5411

-7.4036

-6.8187

-6.5476

-6.6495

-8.8907

176.7147

175.3999

174.9644

174.4784

173.5245

170.0813

9.4087

-1.5671

-3.4681

-4.9715

-7.5061

-19.5694

-170.9099

-179.2939

177.9581

175.2531

168.3990

32.5350

5.5284

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 236
Table A. 1-3 (cont.).

.300 -26.9084 1.8071

.400 -26.4214 -.2921

.500 -31.3524 -3.7723

.600 -32.5226 -173.9667

.700 -19.4570 -179.1498

.800 -13.1654 179.5663

.900 -8.9631 178.7519

1.000 -5.9232 178.0840

1.100 -3.6844 177.4714

1.200 -2.0636 176.8753

1.300 -.9527 176.2746

1.400 -.2815 175.6563

1.500 .0000 175.0112

1.600 -.0698 174.3334

1.700 -.4583 173.6198

1.800 - 1.1352 172.8708

1.900 -2.0696 172.0919

2.000 -3.2284 171.2945

2.100 -4.5757 170.4970

2.200 -6.0750 169.7234

2.300 -7.6927 168.9980

2.400 -9.4049 168.3375

2.500 -11.2056 167.7404

2.600 -13.1118 167.1799

2.700 -15.1655 166.6032

2.800 -17.4292 165.9359

2.900 -19.9777 165.0876

3.000 -22.8822 163.9546

3.100 -26.1829 162.4335

3.200 -29.8255 160.4955

3.300 -33.5347 158.4514

3.400 -36.6811 157.3768

3.500 -38.5219 158.4827

3.600 -39.0151 161.0152

3.700 -38.9999 163.1452

3.800 -39.3983 164.2240

3.900 -40.8872 164.3876

4.000 -44.2708 163.6748

4.100 -52.1304 160.4922

4.200 -58.9859 -4.9786

4.300 -48.4672 -13.0130

4.400 -46.1446 -14.6340

4.500 -47.1148 -15.5939

4.600 -52.0485 -16.1098

4.700 -76.4941 148.1520

4.800 -51.3888 160.6499

4.900 -47.1369 160.1774

5.000 .-46.4112 159.5018

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 237
A.2 SCANRAP INPUT AND OUTPUT

The SCANned Reflector Analysis Program (SCANRAP) was discussed in Sec. 5.2.

SCANRAP is used to compute the excitations for the array feed elements for the

axisymmetric reflector example in Sec. A.I. It is desired to electronically scan the

reflector to Os =, 1.5 °. The reflector is a smooth axisymmetric reflector with diameter

D = 100,t and F/D-_ 0.4. The feed is a seven element linear array with A/2-spacing

located symmetrically about the focus in the focal plane. The feed elements have cos_0

field pattern with q--- 3.0. The input t'tle for the analysis is shown in Table A.2-1. The

output file for the SCANRAP analysis is shown in Table A.2-2.

9. APPENDIX A: EXAMPLE COMPUTER INPUTS AND OUTPUTS 238
TableA.2-1.

SCANRAP input file for an axisymmetric reflector with a seven element linear array

feed. Refer to Tabs. 5.2-1 and 5.2-2 for the format and variable descriptions. The direction of scan is 0s = 1.5".

7

1.5000 40.0000 0.0000

1.0000 40.0000 0.0000

.5000 40.0000 0.0000

.0000 40.0000 0.0000

-.5000 40.0000 0.0000

- 1.0000 40.0000 0.0000

- 1.5000 40.0000 0.0000

3.0000

100.0000 0.0000 40.0000

1.50

20

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 239
Table A.2-2.

SCANRAP output file for an axisymmetric reflector with a seven element linear array

feed. The direction of scan is 0s-- 1.5 °.

SCANRAP, VERSION 1.0, NOVEMBER 17, 1988

**********FEED DATA**********

THERE ARE 7 FEEDS

FEED X-COORD Z-COORD TFEED

1 1.5000 40.0000 .0000

2 1.0000 40.0000 .0000

3 .5OO0 40.OOOO .0000

4 .0000 40.0000 .0000

5 -.5000 40.0000 .0000

6 - 1.0000 40.0000 .0000

7 -1.5000 40.0000 .0000

THE FEED COSINE EXPONENT -- 3.0000

**********REFLECTOR DATA**********

REFLECTOR DIAMETER (WAVELENGTHS) -- 100.0000

REFLECTOR OFFSET (WAVELENGTHS) = .0000

REFLECTOR FOCAL LENGTH (WAVELENGTHS) = 40.0000

REFLECTOR F/DP = .40000

**********SCAN ANGLE**********

THE SCAN ANGLE (DEG) = 1.5000

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 240
Table A.2-2 (cont.).

**********INTEGRATION DATA**********

THE QUADRATURE IS ORDER 20

ZERO WEIGHT

-.99313E+00 .17614E.01

-.96397E + 00 .40601E-01

-.91223E + 00 .62672E-01

-.83912E + 00 .83277E.01

-.74633E + 00 . I0193E + 00

-.63605E +00 .11819E+00

-.51087E + 00 .13169E+ 00

-.37371E+ 00 .14210E+00

-.22779E + 00 .14917E + 00

-.76527E-01 .15275E + 00

.76527E-01 .15275E + 00

.22779E + 00 .14917E + 00

.37371E + 00 .14210E + O0

.51087E + 00 .13169E + 00

.63605E + 00 .11819E+ 00

.74633E+00 .10193E+00

.83912E + 00 .83277E-01

.91223 E + 00 .62672 E-01

.96397E + 00 .40601E-01

.99313E+00 .17614E-01

**************************

FEED EXCITATION

FEED AMP

(LINEAR)

PHASE X-COORD Z-COORD

(DEG)

1 .30003E-01 179.8680 1.5000 40.0000

2 .41161E-OI -1.8880 1.0000 40.0000

3 .51191E-01 177.2723 .5000 40.0000

4 .59228E-01 -2.8996 .0000 40.0000

5 .56011E-01 -.5240 -.5000 40.0000

6 . 10000E + 01 .0000 - 1.0000 40.0000

7 .83784E+00 .3778 -1.5000 40.0000

TFEED

.0000

.0000

.0000

.0000

.0000

.0000

.0000

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 241
A.3 DRAPCA INPUT AND OUTPUT

The Dual Reflector Analysis Program for Cylindrical Antennas (DRAPCA) was

discussed in Sec. 5.3. An example of its use is presented in this section. DRAPCA is

used to compute a pattern for an axisymmetric dual cylindrical reflector. The geometry

for the reflector is shown in Fig. 6.1-6. The pattern is scanned by displacing the feed

from the focus. This example is the scanned pattern shown in Fig. 6.4-2 (smooth

reflector).

The feed is displaced by 9,1 in the minus x-direction and by 4.5,t in the plus

z-direction. The feed is tilted up by 10". The input file for the analysis is shown in Table

A.3-1. The output file for the DRAPCA analysis is shown in Table A.3-2.

9. APPENDIX A: EXAMPLE COMPUTER INPUTS AND OUTPUTS 242
Table A.3-1.

DRAPCA input file for the axisymmetric dual cylindrical reflector of Fig. 6.1-6. Refer

to Tabs. 5.3-1 and 5.3-2 for the format and variable descriptions. The direction of scan

is Os = 5.7 ° •

0

1 2

1.0000 0.0000

10.0000 0.0000

100.0000 0.0000

34.0000 3.0000

0.0000 10.0000

20

-9.0000 30.5000 10.0000

0.0000

50.0000

24.0000 0.0000 0.0000

0.1000

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 243
Table A.3-2.

DRAPCA output f'de for the axisymmetric dual cylindrical reflector of Fig. 6.1-6.

direction of scan is Os-- 5.7*.

TheDRAPCA, VERSION 1.0, JANUARY 21,1990

**********FEED DATA**********

THERE ARE 1 FEEDS

THE FEED LOCATIONS ARE IN THE PRIMED COORDINATE SYSTEM

THEIR EXCITATIONS (LINEAR AND DEGREES)

AND LOCATIONS (WAVELENGTHS) ARE:

FEED AMP PHASE X-COORD Z-COORD TFEED

1 1.0000 .0000 -9.0000 30.5000 10.0000

THE FEED COSINE EXPONENT -- 10.0000

THE MAX PHASE ERROR = .0000 DEGREES

THE NUMBER OF PERIODS OF THE PHASE ERROR = .0000

**********MAIN REFLECTOR DATA**********

MAIN REFLECTOR DIAMETER (WAVELENGTHS) = 100.0000

MAIN REFLECTOR OFFSET (WAVELENGTHS) = .0000

MAIN REFLECTOR FOCAL LENGTH (WAVELENGTHS) = 50.0000

MAIN REFLECTOR F/DP = .50000

ILLUMINATION AT THE TOP OF THE MAIN REFLECTOR (DB) -- -12.7122

ILLUMINATION AT THE BOTTOM OF THE MAIN REFLECTOR (DB) --

-5.8027

**********SUB REFLECTOR DATA**********

SUBREFLECTOR DIAMETER (WAVELENGTHS) -- 34.0000

SUBREFLECTOR OFFSET (WAVELENGTHS) = .0003

FEED PHASE CENTER OFFSET (WAVELENGTHS) -- .0000

INTERFOCAL DISTANCE (WAVELENGTHS) = 24.0000

SUBREFLECTOR ECCENTRICITY -- 3.0000

EQUIVALENT VIRTUAL FOCAL LENGTH (WAVELENGTHS) =

EQUIVALENT VIRTUAL F/DP -- 1.0000

INTERFOCAL AXIS TILT (DEGREES) = .0000

Fl COORDINATE SYSTEM TILT (DEGREES) - .0000

ILLUMINATION AT THE TOP OF THE SUBREFLECTOR (DB) =

ILLUMINATION AT THE BOTTOM OF THE SUBREFLECTOR (DB) =

100.0000

-17.3176

-5.8308

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 244
Table A.3-2. (cont.)

**********OUTPUT PATTERN**********

THETA MIN = .0000

THETA MAX = 10.0000

THETA INC = .1000

**********INTEGRATION DATA**********

THE QUADRATURE IS ORDER 20

ZERO WEIGHT

-.99313E+00 .17614E-01

-.96397E + O0 .40601E-01

-.91223 E + O0 .62672E-01

-.83912E + 00 .83277E-01

-.74633E + 00 .10193E+ O0

-.63605E + O0 .11819E+00

-.51087E + O0 .13169E + O0

-.37371E + 00 .14210E + O0

-.22779E + 00 .14917E + O0

-.76527E-01 .15275E + O0

.76527E-01 .15275E + O0

.22779E + 00 .14917E + 00

.37371E + O0 .142 IOE + O0

.51087E+00 .13169E+00

.63605E + O0 .11819E + 00

.74633E + O0 .10193E + 00

.83912E + O0 .83277E-01

.91223E + O0 .62672E-01

.96397E + 00 .40601E-01

.99313E+ 00 .17614E-01

THE MAXIMUM PATTERN VALUE IS .28422E+00 AT 5.7000 DEGREES

ANGLE (DEG) MAGNITUDE (DB) PHASE(DEG)

.000 -31.9438 -18.9909

.100 -33.4471 -34.2327

.200 -37.3744 -65.3321

.300 -38.1297 -133.8692

.400 -33.4901 -172.6599

.500 -31.0264 170.3503

.600 -30.8535 157.1399

.700 -32.9799 139.5007

.800 -37.1144 98.2908

.900 -35.4026 31.0677

1.000 -31.1502 1.4702

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 245
Table A.3-2. (cont.)

1.100 -29.3102 - 13.7232

1.200 -29.6899 -27.4507

1.300 -32.4222 -48.7230

1.400 -35.9800 - 101.7306

1.500 -32.3844 -160.1923

1.600 -28.7308 176.2502

1.700 -27.4486 162.0109

1.800 -28.3462 147.2063

1.900 -31.5846 120.7678

2.000 -33.6274 58.4323

2.100 -29.1714 11.3999

2.200 -26.1020 -8.4003

2.300 -25.2881 -22.2649

2.400 -26.6288 -38.6047

2.500 -30.1238 -71.3956

2.600 -30.2574 -136.9776

2.700 -25.6991 - 174.6383

2.800 -23.0753 167.8565

2.900 -22.5825 154.1416

3.000 -24. 2049 136. 4060

3.100 -27.6440 98.2948

3.200 -26.3505 34.0248

3.300 -21.8655 1.8649

3.400 -19.4118 -14.1400

3.500 - 18.9634 -27.2309

3.600 -20.5522 -44.4493

3.700 -23.9544 -81. 8968

3.800 -22.5893 - 146.4935

3.900 - 17. 8227 - 178.4675

4.000 -15.0170 166.3728

4. lO0 - 14.0875 155.0060

4.200 -14.9926 142.0730

4.300 - 18.0966 118.1600

4.400 -20.3602 56.2266

4.500 - 15.0781 6.9875

4.600 -i0.8693 -! 1.2422

4.700 -8.5954 -21.5467

4.800 -7.8371 -29.8961

4.900 -8.5675 -38.9930

5.000 -11.2262 -53.3059

5.100 -16.3169 -93.7612

5.200 - 13.5250 - 170.5952

5.300 -7.5566 163.0560

5.400 -3.9387 151.9990

5.500 -1.7509 144.7387

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 246
Table A.3-2. (cont.)

5.600 -.5155 138.7667

5.700 .0000 133.2159

5.800 -.0731 127.6729

5.900 -.6467 121.8964

6.000 -1.6477 115.7486

6.100 -3.0013 109.2123

6.200 -4.6215 102.4612

6.300 -6.4116 95.9413

6.400 -8.2733 90.3684

6.500 -10.1181 86.5111

6.600 - 11.8727 84.7364

6.700 - 13.5000 84.5438

6.800 - 15.0529 84.5123

6.900 - 16.7036 82.7866

7.000 - 18.6791 77.4579

7.100 -21.1105 66.2449

7.200 -23.7309 46.3416

7.300 -25.5969 18.4428

7.400 -26.3587 -8.3027

7.500 -27.2863 -28.3221

7.600 -29.7146 -44.7486

7.700 -35.2893 -67.2161

7.800 -41.8928 - 160.7856

7.900 -33.9064 144.0989

8.000 -30.7186 128.2787

8.1 O0 -30.4240 115.9502

8.200 -32.5034 99.7713

8.300 -36.8574 64.4989

8.400 -36.9405 -3.7418

8.500 -32.8848 -38.73 I0

8.600 -31.0580 -55.3581

8.700 -31.4204 -68.8361

8.800 -34.0282 -86.7004

8.900 -38.7078 - 127.6477

9.000 -37.6045 163.5279

9.100 -33.6837 132.8955

9.200 -32.1658 117.3623

9.300 -32.8236 103.8998

9.400 -35.7792 84.7363

9.500 -40.4364 38.0176

9.600 -38.2254 -27.7945

9.700 -34.5773 -54.5917

9.800 -33.3671 -68.9905

9.900 -34.3358 -82.0178

10.000 -37.7479 - 101.7240

APPENDIX A. EXAMPLE COMPUTER INPUTS AND OUTPUTS 247